| L(s) = 1 | − 2·3-s − 2·5-s + 9-s − 4·11-s + 2·13-s + 4·15-s + 2·17-s − 25-s + 4·27-s − 8·29-s + 8·31-s + 8·33-s − 4·37-s − 4·39-s + 10·41-s − 2·43-s − 2·45-s − 4·51-s + 8·53-s + 8·55-s − 14·59-s − 4·65-s + 8·67-s − 8·71-s − 6·73-s + 2·75-s + 79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s − 1/5·25-s + 0.769·27-s − 1.48·29-s + 1.43·31-s + 1.39·33-s − 0.657·37-s − 0.640·39-s + 1.56·41-s − 0.304·43-s − 0.298·45-s − 0.560·51-s + 1.09·53-s + 1.07·55-s − 1.82·59-s − 0.496·65-s + 0.977·67-s − 0.949·71-s − 0.702·73-s + 0.230·75-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.29104885406487, −12.69688921956807, −12.28029108819278, −11.92290059351762, −11.45286637984063, −11.05117483725539, −10.68795827772862, −10.30209496424253, −9.706118153050302, −9.182509903451765, −8.487248014218732, −8.092980463509502, −7.703966722784062, −7.201492578228133, −6.679150009008556, −6.046809264889094, −5.627288727864931, −5.325056147916954, −4.693873884484557, −4.121050325271983, −3.752335052106422, −2.863145565562501, −2.632813881310204, −1.551103449493818, −0.9942245813994134, 0, 0,
0.9942245813994134, 1.551103449493818, 2.632813881310204, 2.863145565562501, 3.752335052106422, 4.121050325271983, 4.693873884484557, 5.325056147916954, 5.627288727864931, 6.046809264889094, 6.679150009008556, 7.201492578228133, 7.703966722784062, 8.092980463509502, 8.487248014218732, 9.182509903451765, 9.706118153050302, 10.30209496424253, 10.68795827772862, 11.05117483725539, 11.45286637984063, 11.92290059351762, 12.28029108819278, 12.69688921956807, 13.29104885406487
