| L(s) = 1 | + 3-s − 3.20·5-s − 3.68·7-s + 9-s − 2.41·11-s − 0.292·13-s − 3.20·15-s + 7.19·17-s + 2.96·19-s − 3.68·21-s − 5.50·23-s + 5.28·25-s + 27-s + 3.32·29-s + 3.32·31-s − 2.41·33-s + 11.8·35-s − 0.357·37-s − 0.292·39-s + 3.49·41-s + 11.1·43-s − 3.20·45-s + 0.406·47-s + 6.59·49-s + 7.19·51-s − 10.6·53-s + 7.73·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.43·5-s − 1.39·7-s + 0.333·9-s − 0.727·11-s − 0.0810·13-s − 0.828·15-s + 1.74·17-s + 0.680·19-s − 0.804·21-s − 1.14·23-s + 1.05·25-s + 0.192·27-s + 0.616·29-s + 0.597·31-s − 0.419·33-s + 1.99·35-s − 0.0587·37-s − 0.0468·39-s + 0.546·41-s + 1.69·43-s − 0.478·45-s + 0.0593·47-s + 0.942·49-s + 1.00·51-s − 1.46·53-s + 1.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 0.292T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + 0.357T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 0.406T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 + 4.18T + 67T^{2} \) |
| 71 | \( 1 + 2.68T + 71T^{2} \) |
| 73 | \( 1 + 6.27T + 73T^{2} \) |
| 79 | \( 1 + 6.66T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.51T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70909221444122392259669913202, −7.02885016568589479083637620048, −6.13269021928201127199119163210, −5.44941518726886528733752112378, −4.39443460744983577897792250520, −3.76447512690177841282856997774, −3.13841879255035874565313779878, −2.64053315201547211358656128563, −1.04699263015907031249581359895, 0,
1.04699263015907031249581359895, 2.64053315201547211358656128563, 3.13841879255035874565313779878, 3.76447512690177841282856997774, 4.39443460744983577897792250520, 5.44941518726886528733752112378, 6.13269021928201127199119163210, 7.02885016568589479083637620048, 7.70909221444122392259669913202
