| L(s) = 1 | − 2·3-s + 4-s + 9-s + 18·11-s − 2·12-s − 8·17-s − 12·19-s − 8·23-s + 12·25-s + 2·27-s + 14·29-s − 36·33-s + 36-s + 6·37-s − 12·41-s − 14·43-s + 18·44-s + 49-s + 16·51-s − 12·53-s + 24·57-s + 24·59-s − 4·61-s − 64-s − 8·68-s + 16·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s + 5.42·11-s − 0.577·12-s − 1.94·17-s − 2.75·19-s − 1.66·23-s + 12/5·25-s + 0.384·27-s + 2.59·29-s − 6.26·33-s + 1/6·36-s + 0.986·37-s − 1.87·41-s − 2.13·43-s + 2.71·44-s + 1/7·49-s + 2.24·51-s − 1.64·53-s + 3.17·57-s + 3.12·59-s − 0.512·61-s − 1/8·64-s − 0.970·68-s + 1.92·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215831862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215831862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 89 T^{2} + 492 T^{3} + 2232 T^{4} + 492 p T^{5} + 89 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 14 T^{2} + 32 T^{3} + 499 T^{4} + 32 p T^{5} + 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 14 T + 101 T^{2} - 518 T^{3} + 2500 T^{4} - 518 p T^{5} + 101 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2778 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T + 64 T^{2} - 312 T^{3} + 1779 T^{4} - 312 p T^{5} + 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 117 T^{2} + 828 T^{3} + 5048 T^{4} + 828 p T^{5} + 117 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 64 T^{2} + 644 T^{3} + 7147 T^{4} + 644 p T^{5} + 64 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 4059 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 6 p T^{2} - 3888 T^{3} + 34091 T^{4} - 3888 p T^{5} + 6 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 107 T^{2} + 4 T^{3} + 10432 T^{4} + 4 p T^{5} - 107 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 76 T^{2} + 384 T^{3} - 93 T^{4} + 384 p T^{5} + 76 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 27366 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 14090 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 48 T + 1129 T^{2} + 17328 T^{3} + 190752 T^{4} + 17328 p T^{5} + 1129 p^{2} T^{6} + 48 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 18 T + 248 T^{2} + 2520 T^{3} + 20667 T^{4} + 2520 p T^{5} + 248 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.967288604788083818557224881256, −7.10273666132993184357042529551, −7.06230048433254122061983068154, −6.96424564403085186636217068398, −6.59729671185748879788068475144, −6.52543997429274100138818482948, −6.46706840243440355843281942084, −6.21537209994820167311612232207, −6.21264242957605226308921733799, −5.90577736808479048162270825976, −5.21206160125379937896494216959, −4.97470069741193896304403926685, −4.68965757326708442835757912045, −4.42392860858712601916531713415, −4.41470713949315019862199537618, −3.92301951337008298282573745801, −3.80530960900091386705203995356, −3.62903464659131082213049936290, −3.06389081997465632949987871796, −2.63938249488714978254864761019, −2.22309253777135583377504159053, −1.74365207528250797515571452263, −1.60261328394993151849986562835, −1.14369903608284725194696881675, −0.55780402277734569359906106642,
0.55780402277734569359906106642, 1.14369903608284725194696881675, 1.60261328394993151849986562835, 1.74365207528250797515571452263, 2.22309253777135583377504159053, 2.63938249488714978254864761019, 3.06389081997465632949987871796, 3.62903464659131082213049936290, 3.80530960900091386705203995356, 3.92301951337008298282573745801, 4.41470713949315019862199537618, 4.42392860858712601916531713415, 4.68965757326708442835757912045, 4.97470069741193896304403926685, 5.21206160125379937896494216959, 5.90577736808479048162270825976, 6.21264242957605226308921733799, 6.21537209994820167311612232207, 6.46706840243440355843281942084, 6.52543997429274100138818482948, 6.59729671185748879788068475144, 6.96424564403085186636217068398, 7.06230048433254122061983068154, 7.10273666132993184357042529551, 7.967288604788083818557224881256
