| L(s) = 1 | − 2.87·3-s − 5-s + 5.29·9-s + 1.46·11-s − 2.22·13-s + 2.87·15-s − 7.26·17-s + 5.48·19-s + 2.51·23-s + 25-s − 6.60·27-s − 7.12·29-s + 6.51·31-s − 4.22·33-s + 6.90·37-s + 6.39·39-s + 11.3·41-s + 3.31·43-s − 5.29·45-s + 8.36·47-s + 20.9·51-s − 7.00·53-s − 1.46·55-s − 15.8·57-s − 9.07·59-s − 11.2·61-s + 2.22·65-s + ⋯ |
| L(s) = 1 | − 1.66·3-s − 0.447·5-s + 1.76·9-s + 0.441·11-s − 0.616·13-s + 0.743·15-s − 1.76·17-s + 1.25·19-s + 0.524·23-s + 0.200·25-s − 1.27·27-s − 1.32·29-s + 1.17·31-s − 0.734·33-s + 1.13·37-s + 1.02·39-s + 1.76·41-s + 0.505·43-s − 0.789·45-s + 1.22·47-s + 2.93·51-s − 0.962·53-s − 0.197·55-s − 2.09·57-s − 1.18·59-s − 1.43·61-s + 0.275·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.87T + 3T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 7.26T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 + 9.07T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
| 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
−9.038994738944160403881093238392, −7.64342307956808619007463142059, −7.17529724876794674459621387397, −6.26988213219476380348217427456, −5.69899876905843456346627494237, −4.62236045265392216322657400640, −4.28754350583900050823602510376, −2.75883715249086266604990613246, −1.21486236431899542509165270072, 0,
1.21486236431899542509165270072, 2.75883715249086266604990613246, 4.28754350583900050823602510376, 4.62236045265392216322657400640, 5.69899876905843456346627494237, 6.26988213219476380348217427456, 7.17529724876794674459621387397, 7.64342307956808619007463142059, 9.038994738944160403881093238392
