| L(s) = 1 | − 2-s + 0.430·3-s + 4-s + 1.34·5-s − 0.430·6-s − 7-s − 8-s − 2.81·9-s − 1.34·10-s + 4.88·11-s + 0.430·12-s + 4.15·13-s + 14-s + 0.577·15-s + 16-s + 4.03·17-s + 2.81·18-s − 0.649·19-s + 1.34·20-s − 0.430·21-s − 4.88·22-s − 2.19·23-s − 0.430·24-s − 3.20·25-s − 4.15·26-s − 2.50·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.248·3-s + 0.5·4-s + 0.599·5-s − 0.175·6-s − 0.377·7-s − 0.353·8-s − 0.938·9-s − 0.423·10-s + 1.47·11-s + 0.124·12-s + 1.15·13-s + 0.267·14-s + 0.149·15-s + 0.250·16-s + 0.979·17-s + 0.663·18-s − 0.149·19-s + 0.299·20-s − 0.0940·21-s − 1.04·22-s − 0.458·23-s − 0.0879·24-s − 0.640·25-s − 0.815·26-s − 0.482·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.866600609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866600609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 0.430T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 + 0.649T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 0.741T + 29T^{2} \) |
| 31 | \( 1 + 0.702T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 + 9.84T + 53T^{2} \) |
| 59 | \( 1 + 0.936T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 + 0.0975T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
| 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
−8.103166612580033528788019314569, −7.59311791950596111905378961424, −6.51192282587690986406644438544, −6.02738088181478887621628577432, −5.66472881885965742131490624469, −4.19144641826844715609461597900, −3.53711832634225580391879081816, −2.68289093097309813103158581088, −1.70506449092377827771532945809, −0.824166769725427444514092361864,
0.824166769725427444514092361864, 1.70506449092377827771532945809, 2.68289093097309813103158581088, 3.53711832634225580391879081816, 4.19144641826844715609461597900, 5.66472881885965742131490624469, 6.02738088181478887621628577432, 6.51192282587690986406644438544, 7.59311791950596111905378961424, 8.103166612580033528788019314569
