| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 2·13-s − 4·14-s + 16-s − 2·17-s + 4·19-s − 2·23-s − 5·25-s + 2·26-s + 4·28-s − 2·29-s + 4·31-s − 32-s + 2·34-s + 6·37-s − 4·38-s − 6·41-s + 12·43-s + 2·46-s − 4·47-s + 9·49-s + 5·50-s − 2·52-s + 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.417·23-s − 25-s + 0.392·26-s + 0.755·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s − 0.937·41-s + 1.82·43-s + 0.294·46-s − 0.583·47-s + 9/7·49-s + 0.707·50-s − 0.277·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 4091 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.46503026150184, −13.83586999464527, −13.50605383973779, −12.74410112129425, −12.11866777960828, −11.62069718811032, −11.43576940217826, −10.86589170125580, −10.20314624657067, −9.873977505217610, −9.185148626116638, −8.797477826780252, −8.036408294404723, −7.836382395227244, −7.366744107156154, −6.729099279388959, −6.003982799366524, −5.467885740277832, −4.929461665788720, −4.288532578690780, −3.764910046066690, −2.722333003776997, −2.302467519087900, −1.571331703016137, −1.002061385356564, 0,
1.002061385356564, 1.571331703016137, 2.302467519087900, 2.722333003776997, 3.764910046066690, 4.288532578690780, 4.929461665788720, 5.467885740277832, 6.003982799366524, 6.729099279388959, 7.366744107156154, 7.836382395227244, 8.036408294404723, 8.797477826780252, 9.185148626116638, 9.873977505217610, 10.20314624657067, 10.86589170125580, 11.43576940217826, 11.62069718811032, 12.11866777960828, 12.74410112129425, 13.50605383973779, 13.83586999464527, 14.46503026150184
