| L(s) = 1 | − 2-s − 1.30·3-s + 4-s − 2.30·5-s + 1.30·6-s − 2.60·7-s − 8-s − 1.30·9-s + 2.30·10-s − 11-s − 1.30·12-s − 4·13-s + 2.60·14-s + 3·15-s + 16-s + 0.697·17-s + 1.30·18-s − 0.302·19-s − 2.30·20-s + 3.39·21-s + 22-s − 2.30·23-s + 1.30·24-s + 0.302·25-s + 4·26-s + 5.60·27-s − 2.60·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.752·3-s + 0.5·4-s − 1.02·5-s + 0.531·6-s − 0.984·7-s − 0.353·8-s − 0.434·9-s + 0.728·10-s − 0.301·11-s − 0.376·12-s − 1.10·13-s + 0.696·14-s + 0.774·15-s + 0.250·16-s + 0.169·17-s + 0.307·18-s − 0.0694·19-s − 0.514·20-s + 0.740·21-s + 0.213·22-s − 0.480·23-s + 0.265·24-s + 0.0605·25-s + 0.784·26-s + 1.07·27-s − 0.492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2716535232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2716535232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 0.697T + 17T^{2} \) |
| 19 | \( 1 + 0.302T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 0.0916T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
| 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
−10.13908254638491083874951990210, −9.269034111763925044307005793118, −8.328799290077530671402954717796, −7.53344889284682804787513681027, −6.75869715455989844105356081784, −5.88927829967886162992233096481, −4.86664397093255908871564549420, −3.59814351494281744144529152380, −2.54810285673487757348618055601, −0.44138953717942556569821409933,
0.44138953717942556569821409933, 2.54810285673487757348618055601, 3.59814351494281744144529152380, 4.86664397093255908871564549420, 5.88927829967886162992233096481, 6.75869715455989844105356081784, 7.53344889284682804787513681027, 8.328799290077530671402954717796, 9.269034111763925044307005793118, 10.13908254638491083874951990210
