| L(s) = 1 | + 2-s + 3-s + 4-s + 4.10·5-s + 6-s − 1.42·7-s + 8-s + 9-s + 4.10·10-s − 11-s + 12-s + 1.35·13-s − 1.42·14-s + 4.10·15-s + 16-s + 3.83·17-s + 18-s − 3.67·19-s + 4.10·20-s − 1.42·21-s − 22-s + 2.60·23-s + 24-s + 11.8·25-s + 1.35·26-s + 27-s − 1.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.83·5-s + 0.408·6-s − 0.537·7-s + 0.353·8-s + 0.333·9-s + 1.29·10-s − 0.301·11-s + 0.288·12-s + 0.376·13-s − 0.380·14-s + 1.06·15-s + 0.250·16-s + 0.929·17-s + 0.235·18-s − 0.843·19-s + 0.918·20-s − 0.310·21-s − 0.213·22-s + 0.543·23-s + 0.204·24-s + 2.37·25-s + 0.266·26-s + 0.192·27-s − 0.268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.311743245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.311743245\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 0.832T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 + 7.11T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.591649731779280139715795430172, −7.59868658601754921410085778074, −6.66354441500548979521198009748, −6.24678819998694537140557624100, −5.45295048688876914529769771965, −4.86196392080172331094765225727, −3.67048425041975583588039960510, −2.90522371859641483678065134706, −2.18344814520338804248528388513, −1.29647763776162242271983671926,
1.29647763776162242271983671926, 2.18344814520338804248528388513, 2.90522371859641483678065134706, 3.67048425041975583588039960510, 4.86196392080172331094765225727, 5.45295048688876914529769771965, 6.24678819998694537140557624100, 6.66354441500548979521198009748, 7.59868658601754921410085778074, 8.591649731779280139715795430172
