| L(s) = 1 | + 2.06·2-s − 3.71·3-s − 3.72·4-s + 5·5-s − 7.67·6-s − 13.2·7-s − 24.2·8-s − 13.2·9-s + 10.3·10-s − 11·11-s + 13.8·12-s − 89.1·13-s − 27.3·14-s − 18.5·15-s − 20.3·16-s − 50.4·17-s − 27.3·18-s + 19·19-s − 18.6·20-s + 49.0·21-s − 22.7·22-s − 14.5·23-s + 89.9·24-s + 25·25-s − 184.·26-s + 149.·27-s + 49.2·28-s + ⋯ |
| L(s) = 1 | + 0.731·2-s − 0.714·3-s − 0.465·4-s + 0.447·5-s − 0.522·6-s − 0.714·7-s − 1.07·8-s − 0.490·9-s + 0.327·10-s − 0.301·11-s + 0.332·12-s − 1.90·13-s − 0.522·14-s − 0.319·15-s − 0.318·16-s − 0.720·17-s − 0.358·18-s + 0.229·19-s − 0.208·20-s + 0.510·21-s − 0.220·22-s − 0.131·23-s + 0.765·24-s + 0.200·25-s − 1.39·26-s + 1.06·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5532546366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5532546366\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 2.06T + 8T^{2} \) |
| 3 | \( 1 + 3.71T + 27T^{2} \) |
| 7 | \( 1 + 13.2T + 343T^{2} \) |
| 13 | \( 1 + 89.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 14.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 82.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 64.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 178.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 409.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 109.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 744.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 982.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 717.T + 9.12e5T^{2} \) |
| 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
−9.556213504032750112874567585830, −8.972296413918627262326870985823, −7.74921514902749114136857071873, −6.66423797298725915508565140873, −5.96138561238066883943866856123, −5.14338669304066922834840408150, −4.60105767163017414205524845505, −3.25016825868163039578670224748, −2.38761969914567514095379097987, −0.34057326839536413674306655113,
0.34057326839536413674306655113, 2.38761969914567514095379097987, 3.25016825868163039578670224748, 4.60105767163017414205524845505, 5.14338669304066922834840408150, 5.96138561238066883943866856123, 6.66423797298725915508565140873, 7.74921514902749114136857071873, 8.972296413918627262326870985823, 9.556213504032750112874567585830
