| L(s) = 1 | − 1.50·3-s − 2.01·5-s + 3.94·7-s − 0.721·9-s − 0.829·11-s + 2.55·13-s + 3.03·15-s + 0.308·17-s − 19-s − 5.96·21-s + 2.11·23-s − 0.956·25-s + 5.61·27-s − 1.91·29-s − 8.33·31-s + 1.25·33-s − 7.93·35-s − 0.387·37-s − 3.85·39-s + 3.22·41-s − 1.93·43-s + 1.45·45-s − 11.5·47-s + 8.59·49-s − 0.465·51-s − 5.93·53-s + 1.66·55-s + ⋯ |
| L(s) = 1 | − 0.871·3-s − 0.899·5-s + 1.49·7-s − 0.240·9-s − 0.250·11-s + 0.709·13-s + 0.783·15-s + 0.0747·17-s − 0.229·19-s − 1.30·21-s + 0.441·23-s − 0.191·25-s + 1.08·27-s − 0.355·29-s − 1.49·31-s + 0.218·33-s − 1.34·35-s − 0.0637·37-s − 0.617·39-s + 0.504·41-s − 0.295·43-s + 0.216·45-s − 1.68·47-s + 1.22·49-s − 0.0651·51-s − 0.815·53-s + 0.224·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 + 0.829T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 0.308T + 17T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 8.33T + 31T^{2} \) |
| 37 | \( 1 + 0.387T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 + 1.72T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
| 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
−7.907386905660260853198290607980, −7.03184489286033984163548846230, −6.27544448321983493441098255782, −5.31551474329874035961247630906, −5.07807415815692718758026325456, −4.12063560566279774415838244231, −3.44277960829262034830421311860, −2.16693806949069635231078531526, −1.16474553904324311852461178920, 0,
1.16474553904324311852461178920, 2.16693806949069635231078531526, 3.44277960829262034830421311860, 4.12063560566279774415838244231, 5.07807415815692718758026325456, 5.31551474329874035961247630906, 6.27544448321983493441098255782, 7.03184489286033984163548846230, 7.907386905660260853198290607980
