| L(s) = 1 | − 340·4-s − 4.51e4·9-s + 1.09e5·11-s + 2.90e5·16-s + 6.36e5·19-s + 3.53e6·29-s − 1.05e7·31-s + 1.53e7·36-s − 1.67e7·41-s − 3.73e7·44-s − 5.72e7·49-s + 4.60e8·59-s + 3.60e8·61-s − 2.47e8·64-s − 4.76e7·71-s − 2.16e8·76-s + 7.28e8·79-s + 9.91e8·81-s + 1.58e9·89-s − 4.96e9·99-s + 2.27e9·101-s + 2.13e9·109-s − 1.20e9·116-s − 1.29e9·121-s + 3.59e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.664·4-s − 2.29·9-s + 2.26·11-s + 1.10·16-s + 1.12·19-s + 0.927·29-s − 2.05·31-s + 1.52·36-s − 0.927·41-s − 1.50·44-s − 1.41·49-s + 4.95·59-s + 3.33·61-s − 1.84·64-s − 0.222·71-s − 0.744·76-s + 2.10·79-s + 2.56·81-s + 2.67·89-s − 5.19·99-s + 2.17·101-s + 1.44·109-s − 0.615·116-s − 0.547·121-s + 1.36·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.468708728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.468708728\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 85 p^{2} T^{2} - 2733 p^{6} T^{4} + 85 p^{20} T^{6} + p^{36} T^{8} \) |
| 3 | $C_2^2 \wr C_2$ | \( 1 + 5020 p^{2} T^{2} + 12953638 p^{4} T^{4} + 5020 p^{20} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 166900 p^{3} T^{2} + 1461117678198 p^{4} T^{4} + 166900 p^{21} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 54984 T + 5180465446 T^{2} - 54984 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} + 35613791860 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} + 285780369220 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16760 p T + 505756418358 T^{2} - 16760 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} + 5779790962540 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1765860 T + 26950935551038 T^{2} - 1765860 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5293856 T + 59464921598526 T^{2} + 5293856 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} + 231603274936660 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8394276 T + 221313076168966 T^{2} + 8394276 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} + 614109141147100 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} + 1368976020813580 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} + 7684297973864980 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 230414520 T + 28555631923987078 T^{2} - 230414520 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 180245284 T + 30154717014478446 T^{2} - 180245284 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - 41160407446058180 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 23805936 T + 85782754020107086 T^{2} + 23805936 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 229489314868712740 T^{2} + \)\(37\!\cdots\!22\)\( p^{2} T^{4} + 229489314868712740 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 364021760 T + 220545463862625438 T^{2} - 364021760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} + 434569632367965820 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 791350380 T + 832192702699668118 T^{2} - 791350380 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} + 2561123777205326980 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.15610365897770019627057957027, −10.95684822845332849911202376981, −10.16147506556882351552248364648, −9.940682443305697095365037104109, −9.724897683071254735838874897170, −9.014290417731505472672622979190, −8.901692686102500228597966877290, −8.663119059137671796555033698025, −8.385249236641532451845914584856, −7.72074024600832567859722683994, −7.47791034286435820783537735162, −6.69266889868255587779286453488, −6.53114627953068973634711016587, −6.08038321086656746131786019789, −5.49401797864743496641527815617, −5.16507864086958572298745209497, −5.01983196255092295802808109446, −3.81577805038084412277911287987, −3.72570333120829825742928472113, −3.49167886186681066585122114571, −2.70317799575119073246020219026, −2.11361997252452257767929230639, −1.40859646137232419231539161961, −0.65143680316026528490807788708, −0.58335120939399006732167704839,
0.58335120939399006732167704839, 0.65143680316026528490807788708, 1.40859646137232419231539161961, 2.11361997252452257767929230639, 2.70317799575119073246020219026, 3.49167886186681066585122114571, 3.72570333120829825742928472113, 3.81577805038084412277911287987, 5.01983196255092295802808109446, 5.16507864086958572298745209497, 5.49401797864743496641527815617, 6.08038321086656746131786019789, 6.53114627953068973634711016587, 6.69266889868255587779286453488, 7.47791034286435820783537735162, 7.72074024600832567859722683994, 8.385249236641532451845914584856, 8.663119059137671796555033698025, 8.901692686102500228597966877290, 9.014290417731505472672622979190, 9.724897683071254735838874897170, 9.940682443305697095365037104109, 10.16147506556882351552248364648, 10.95684822845332849911202376981, 11.15610365897770019627057957027
