| L(s) = 1 | − 2.10·2-s − 2.17·3-s + 2.43·4-s + 2.39·5-s + 4.58·6-s + 1.34·7-s − 0.920·8-s + 1.73·9-s − 5.04·10-s − 2.33·11-s − 5.30·12-s − 13-s − 2.83·14-s − 5.20·15-s − 2.93·16-s + 5.95·17-s − 3.64·18-s − 3.63·19-s + 5.83·20-s − 2.92·21-s + 4.90·22-s − 0.777·23-s + 2.00·24-s + 0.731·25-s + 2.10·26-s + 2.75·27-s + 3.27·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 1.25·3-s + 1.21·4-s + 1.07·5-s + 1.87·6-s + 0.508·7-s − 0.325·8-s + 0.577·9-s − 1.59·10-s − 0.702·11-s − 1.53·12-s − 0.277·13-s − 0.756·14-s − 1.34·15-s − 0.733·16-s + 1.44·17-s − 0.860·18-s − 0.833·19-s + 1.30·20-s − 0.638·21-s + 1.04·22-s − 0.162·23-s + 0.408·24-s + 0.146·25-s + 0.413·26-s + 0.530·27-s + 0.619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5061084588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5061084588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 2.17T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 + 0.777T + 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 + 0.935T + 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 - 0.251T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 + 4.30T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 0.0610T + 89T^{2} \) |
| 97 | \( 1 + 0.767T + 97T^{2} \) |
| 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
−10.94229087658656599929365756000, −10.18573706658726223835498969603, −9.802950567892006567427737933616, −8.532427749189731639342925877916, −7.73115460065774563725434607991, −6.56268255995651109562822887581, −5.71591037701851222261797112642, −4.78662797687500769460929916336, −2.34223715586854073617878238859, −0.925480073843551150116958291249,
0.925480073843551150116958291249, 2.34223715586854073617878238859, 4.78662797687500769460929916336, 5.71591037701851222261797112642, 6.56268255995651109562822887581, 7.73115460065774563725434607991, 8.532427749189731639342925877916, 9.802950567892006567427737933616, 10.18573706658726223835498969603, 10.94229087658656599929365756000
