| L(s) = 1 | − 6.70·2-s − 4.47·3-s + 29.0·4-s + 25·5-s + 30.0·6-s − 87.2·8-s − 61·9-s − 167.·10-s − 62·11-s − 129.·12-s + 67.0·13-s − 111.·15-s + 121.·16-s + 409.·18-s + 361·19-s + 725.·20-s + 415.·22-s + 390.·24-s + 625·25-s − 450·26-s + 635.·27-s + 750.·30-s + 583.·32-s + 277.·33-s − 1.76e3·36-s + 2.64e3·37-s − 2.42e3·38-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.496·3-s + 1.81·4-s + 5-s + 0.833·6-s − 1.36·8-s − 0.753·9-s − 1.67·10-s − 0.512·11-s − 0.900·12-s + 0.396·13-s − 0.496·15-s + 0.472·16-s + 1.26·18-s + 19-s + 1.81·20-s + 0.859·22-s + 0.677·24-s + 25-s − 0.665·26-s + 0.871·27-s + 0.833·30-s + 0.569·32-s + 0.254·33-s − 1.36·36-s + 1.93·37-s − 1.67·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6874299060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6874299060\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 6.70T + 16T^{2} \) |
| 3 | \( 1 + 4.47T + 81T^{2} \) |
| 7 | \( 1 - 2.40e3T^{2} \) |
| 11 | \( 1 + 62T + 1.46e4T^{2} \) |
| 13 | \( 1 - 67.0T + 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.64e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 791.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.54e4T + 8.85e7T^{2} \) |
| 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
−13.18836900876708261982688968689, −11.67142682040378487914843403625, −10.82477619658826124356223081458, −9.898354266982772451834703022771, −9.007168983878526367304391043211, −7.911359142333311285752965927687, −6.55050977467251966660474252264, −5.44855585979411610412198506732, −2.53597628261584386114806763171, −0.882534227291798518873035730630,
0.882534227291798518873035730630, 2.53597628261584386114806763171, 5.44855585979411610412198506732, 6.55050977467251966660474252264, 7.911359142333311285752965927687, 9.007168983878526367304391043211, 9.898354266982772451834703022771, 10.82477619658826124356223081458, 11.67142682040378487914843403625, 13.18836900876708261982688968689
