| L(s) = 1 | + 3-s + 5-s + 1.51·7-s + 9-s + 2.48·11-s + 13-s + 15-s + 5.76·17-s − 6.24·19-s + 1.51·21-s + 4.73·23-s + 25-s + 27-s − 9.52·29-s + 9.28·31-s + 2.48·33-s + 1.51·35-s − 5.76·37-s + 39-s + 5.76·41-s − 0.969·43-s + 45-s + 7.21·47-s − 4.70·49-s + 5.76·51-s + 9.76·53-s + 2.48·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.572·7-s + 0.333·9-s + 0.749·11-s + 0.277·13-s + 0.258·15-s + 1.39·17-s − 1.43·19-s + 0.330·21-s + 0.987·23-s + 0.200·25-s + 0.192·27-s − 1.76·29-s + 1.66·31-s + 0.432·33-s + 0.256·35-s − 0.947·37-s + 0.160·39-s + 0.900·41-s − 0.147·43-s + 0.149·45-s + 1.05·47-s − 0.672·49-s + 0.807·51-s + 1.34·53-s + 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.436483423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436483423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 - 9.28T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 + 0.969T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 0.0605T + 73T^{2} \) |
| 79 | \( 1 + 0.734T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
| 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
−8.127109193650304391002748688553, −7.39073615832872447944593240263, −6.69811259434627992079301707582, −5.90958902217283201567067458144, −5.21839191106919915243990712747, −4.29536030301661989919148691552, −3.66498923915521134055803262253, −2.71603681066344440758412257194, −1.83060212703966014394952468200, −1.01520188927926912795642285002,
1.01520188927926912795642285002, 1.83060212703966014394952468200, 2.71603681066344440758412257194, 3.66498923915521134055803262253, 4.29536030301661989919148691552, 5.21839191106919915243990712747, 5.90958902217283201567067458144, 6.69811259434627992079301707582, 7.39073615832872447944593240263, 8.127109193650304391002748688553
