| L(s) = 1 | + 2-s + 2·3-s + 3·4-s − 24·5-s + 2·6-s + 35·7-s + 19·8-s − 34·9-s − 24·10-s + 55·11-s + 6·12-s − 50·13-s + 35·14-s − 48·15-s + 39·16-s + 222·17-s − 34·18-s + 160·19-s − 72·20-s + 70·21-s + 55·22-s + 54·23-s + 38·24-s + 38·25-s − 50·26-s − 52·27-s + 105·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.384·3-s + 3/8·4-s − 2.14·5-s + 0.136·6-s + 1.88·7-s + 0.839·8-s − 1.25·9-s − 0.758·10-s + 1.50·11-s + 0.144·12-s − 1.06·13-s + 0.668·14-s − 0.826·15-s + 0.609·16-s + 3.16·17-s − 0.445·18-s + 1.93·19-s − 0.804·20-s + 0.727·21-s + 0.533·22-s + 0.489·23-s + 0.323·24-s + 0.303·25-s − 0.377·26-s − 0.370·27-s + 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 11^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 11^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.047196440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.047196440\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - p T )^{5} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T - p T^{2} - 7 p T^{3} + p^{8} T^{5} - 7 p^{7} T^{7} - p^{10} T^{8} - p^{12} T^{9} + p^{15} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 - 2 T + 38 T^{2} - 92 T^{3} + 1543 T^{4} - 2428 T^{5} + 1543 p^{3} T^{6} - 92 p^{6} T^{7} + 38 p^{9} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 24 T + 538 T^{2} + 9398 T^{3} + 130933 T^{4} + 1626932 T^{5} + 130933 p^{3} T^{6} + 9398 p^{6} T^{7} + 538 p^{9} T^{8} + 24 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 50 T + 5619 T^{2} + 233708 T^{3} + 16338480 T^{4} + 543639348 T^{5} + 16338480 p^{3} T^{6} + 233708 p^{6} T^{7} + 5619 p^{9} T^{8} + 50 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 222 T + 31307 T^{2} - 3492220 T^{3} + 322498276 T^{4} - 24446186380 T^{5} + 322498276 p^{3} T^{6} - 3492220 p^{6} T^{7} + 31307 p^{9} T^{8} - 222 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 160 T + 29251 T^{2} - 3370160 T^{3} + 377191526 T^{4} - 31408900960 T^{5} + 377191526 p^{3} T^{6} - 3370160 p^{6} T^{7} + 29251 p^{9} T^{8} - 160 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 54 T + 19980 T^{2} - 1136032 T^{3} + 384879663 T^{4} - 23050286164 T^{5} + 384879663 p^{3} T^{6} - 1136032 p^{6} T^{7} + 19980 p^{9} T^{8} - 54 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 14 T + 88617 T^{2} - 834200 T^{3} + 3587552730 T^{4} - 24253076052 T^{5} + 3587552730 p^{3} T^{6} - 834200 p^{6} T^{7} + 88617 p^{9} T^{8} - 14 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 34 T + 121594 T^{2} + 1976600 T^{3} + 6479233979 T^{4} + 62495641420 T^{5} + 6479233979 p^{3} T^{6} + 1976600 p^{6} T^{7} + 121594 p^{9} T^{8} + 34 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 1044 T + 681694 T^{2} - 298047902 T^{3} + 99369988485 T^{4} - 25174822625188 T^{5} + 99369988485 p^{3} T^{6} - 298047902 p^{6} T^{7} + 681694 p^{9} T^{8} - 1044 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 114 T + 254723 T^{2} + 29478884 T^{3} + 30866976292 T^{4} + 2882916934612 T^{5} + 30866976292 p^{3} T^{6} + 29478884 p^{6} T^{7} + 254723 p^{9} T^{8} + 114 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 672 T + 248891 T^{2} - 52120304 T^{3} + 6252742678 T^{4} - 118690037344 T^{5} + 6252742678 p^{3} T^{6} - 52120304 p^{6} T^{7} + 248891 p^{9} T^{8} - 672 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 292 T + 402889 T^{2} + 80726736 T^{3} + 71080916080 T^{4} + 10882123012888 T^{5} + 71080916080 p^{3} T^{6} + 80726736 p^{6} T^{7} + 402889 p^{9} T^{8} + 292 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 710 T + 490589 T^{2} + 239496832 T^{3} + 123967485150 T^{4} + 50689150354452 T^{5} + 123967485150 p^{3} T^{6} + 239496832 p^{6} T^{7} + 490589 p^{9} T^{8} + 710 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 270 T + 435470 T^{2} - 207882260 T^{3} + 79414616815 T^{4} - 62972583970436 T^{5} + 79414616815 p^{3} T^{6} - 207882260 p^{6} T^{7} + 435470 p^{9} T^{8} - 270 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 138 T + 558191 T^{2} - 20226020 T^{3} + 164620363932 T^{4} + 7544343798364 T^{5} + 164620363932 p^{3} T^{6} - 20226020 p^{6} T^{7} + 558191 p^{9} T^{8} - 138 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1942 T + 2942616 T^{2} - 2839564732 T^{3} + 2284876388211 T^{4} - 1361693275475580 T^{5} + 2284876388211 p^{3} T^{6} - 2839564732 p^{6} T^{7} + 2942616 p^{9} T^{8} - 1942 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 278 T + 1534836 T^{2} + 342139560 T^{3} + 1020084412007 T^{4} + 174746232948356 T^{5} + 1020084412007 p^{3} T^{6} + 342139560 p^{6} T^{7} + 1534836 p^{9} T^{8} + 278 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 338 T + 1178195 T^{2} + 195729044 T^{3} + 644699644500 T^{4} + 65126455001332 T^{5} + 644699644500 p^{3} T^{6} + 195729044 p^{6} T^{7} + 1178195 p^{9} T^{8} + 338 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 576 T + 966507 T^{2} - 274263328 T^{3} + 100142562874 T^{4} + 8750219144640 T^{5} + 100142562874 p^{3} T^{6} - 274263328 p^{6} T^{7} + 966507 p^{9} T^{8} - 576 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 1644 T + 3592623 T^{2} - 3826259120 T^{3} + 4521678058546 T^{4} - 3299900765078280 T^{5} + 4521678058546 p^{3} T^{6} - 3826259120 p^{6} T^{7} + 3592623 p^{9} T^{8} - 1644 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 3656 T + 8308838 T^{2} + 13139662330 T^{3} + 15848162046361 T^{4} + 14967631162122204 T^{5} + 15848162046361 p^{3} T^{6} + 13139662330 p^{6} T^{7} + 8308838 p^{9} T^{8} + 3656 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 692 T + 3499878 T^{2} - 1743873790 T^{3} + 5505415430657 T^{4} - 2124944136659820 T^{5} + 5505415430657 p^{3} T^{6} - 1743873790 p^{6} T^{7} + 3499878 p^{9} T^{8} - 692 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.303030757436324316042379102997, −8.288255405981257210661816043349, −8.280868152478375221494484507864, −7.87591634441707587969806036031, −7.61971972549975202095004905854, −7.59272422741030994383735960735, −7.52973038249746534999327560179, −7.18871196654525042424385530213, −6.81022882439803477383887122447, −6.19071871044673133053573434768, −6.08610877114800903690189110681, −5.44445197962229711554112207399, −5.43898328882929504099683199673, −5.39422902523692911838483731884, −4.74088595925124231840019849796, −4.33307072597305507644799516639, −4.30681468581577511200178943350, −3.81507756619834076736937503965, −3.53538961054135858131291599592, −3.23310482669984967489602321297, −2.82820150248754049519961152560, −2.32316851763944999091484131038, −1.60068783310082441920212696460, −0.944651254519751412762112291892, −0.839941845743360646497552276066,
0.839941845743360646497552276066, 0.944651254519751412762112291892, 1.60068783310082441920212696460, 2.32316851763944999091484131038, 2.82820150248754049519961152560, 3.23310482669984967489602321297, 3.53538961054135858131291599592, 3.81507756619834076736937503965, 4.30681468581577511200178943350, 4.33307072597305507644799516639, 4.74088595925124231840019849796, 5.39422902523692911838483731884, 5.43898328882929504099683199673, 5.44445197962229711554112207399, 6.08610877114800903690189110681, 6.19071871044673133053573434768, 6.81022882439803477383887122447, 7.18871196654525042424385530213, 7.52973038249746534999327560179, 7.59272422741030994383735960735, 7.61971972549975202095004905854, 7.87591634441707587969806036031, 8.280868152478375221494484507864, 8.288255405981257210661816043349, 8.303030757436324316042379102997
