| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 5.12·7-s − 8-s + 9-s − 10-s + 3.12·11-s + 12-s − 5.12·14-s + 15-s + 16-s − 2·17-s − 18-s − 6·19-s + 20-s + 5.12·21-s − 3.12·22-s − 3.12·23-s − 24-s + 25-s + 27-s + 5.12·28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.93·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.941·11-s + 0.288·12-s − 1.36·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.11·21-s − 0.665·22-s − 0.651·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.968·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.631777203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.631777203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 - 9.12T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
| 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
−8.295858778831265648647717397307, −7.76190986503516542735774339304, −6.96850114688887152038128490205, −6.22276615654099260342162773954, −5.32913397120662782787975404364, −4.41941654093257286246307742227, −3.84071302215742548291349741414, −2.30076780547346667120114416079, −1.98041498385356183753731884054, −1.02025477056700170093807526007,
1.02025477056700170093807526007, 1.98041498385356183753731884054, 2.30076780547346667120114416079, 3.84071302215742548291349741414, 4.41941654093257286246307742227, 5.32913397120662782787975404364, 6.22276615654099260342162773954, 6.96850114688887152038128490205, 7.76190986503516542735774339304, 8.295858778831265648647717397307
