| L(s) = 1 | + 2-s − 1.38·3-s + 4-s − 5-s − 1.38·6-s − 4.20·7-s + 8-s − 1.08·9-s − 10-s − 3.68·11-s − 1.38·12-s − 2.53·13-s − 4.20·14-s + 1.38·15-s + 16-s + 2.39·17-s − 1.08·18-s − 6.47·19-s − 20-s + 5.81·21-s − 3.68·22-s − 3.78·23-s − 1.38·24-s + 25-s − 2.53·26-s + 5.65·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.798·3-s + 0.5·4-s − 0.447·5-s − 0.564·6-s − 1.58·7-s + 0.353·8-s − 0.362·9-s − 0.316·10-s − 1.10·11-s − 0.399·12-s − 0.702·13-s − 1.12·14-s + 0.357·15-s + 0.250·16-s + 0.579·17-s − 0.256·18-s − 1.48·19-s − 0.223·20-s + 1.26·21-s − 0.784·22-s − 0.788·23-s − 0.282·24-s + 0.200·25-s − 0.496·26-s + 1.08·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04368098963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04368098963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 1.38T + 3T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 7.09T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 8.40T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 0.172T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.20T + 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
| 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
−7.64594630865817661765994958691, −6.80018639577250990365812745813, −6.35876823470418320591351252809, −5.71960189482933637685831677478, −5.08846524228442777072451571311, −4.37648123441773106948735709465, −3.36586493670521701778884946747, −2.97749223591494584565366013088, −1.96051327776502773574549597901, −0.087352669274063434525840561642,
0.087352669274063434525840561642, 1.96051327776502773574549597901, 2.97749223591494584565366013088, 3.36586493670521701778884946747, 4.37648123441773106948735709465, 5.08846524228442777072451571311, 5.71960189482933637685831677478, 6.35876823470418320591351252809, 6.80018639577250990365812745813, 7.64594630865817661765994958691
