| L(s) = 1 | + 2·2-s − 10.2·3-s + 4·4-s − 20.5·6-s + 8·8-s + 78.4·9-s − 30.8·11-s − 41.0·12-s − 53.3·13-s + 16·16-s + 1.41·17-s + 156.·18-s + 88.3·19-s − 61.7·22-s + 85.0·23-s − 82.1·24-s − 106.·26-s − 527.·27-s − 49.5·29-s − 62.7·31-s + 32·32-s + 317.·33-s + 2.83·34-s + 313.·36-s − 251.·37-s + 176.·38-s + 547.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.97·3-s + 0.5·4-s − 1.39·6-s + 0.353·8-s + 2.90·9-s − 0.846·11-s − 0.987·12-s − 1.13·13-s + 0.250·16-s + 0.0202·17-s + 2.05·18-s + 1.06·19-s − 0.598·22-s + 0.771·23-s − 0.698·24-s − 0.804·26-s − 3.76·27-s − 0.317·29-s − 0.363·31-s + 0.176·32-s + 1.67·33-s + 0.0143·34-s + 1.45·36-s − 1.11·37-s + 0.754·38-s + 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 10.2T + 27T^{2} \) |
| 11 | \( 1 + 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.41T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 85.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 93.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 211.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 384.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 114.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 135.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 184.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
| 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
−7.59071503795007360547261420806, −7.32427127210245724502068790776, −6.46801161469179809986960092512, −5.62712514235505167104245614054, −5.12758042892308670012906404035, −4.65054718541626820691555921517, −3.52937480511297748516161487033, −2.22979463158591224986333623576, −1.01922074306095030425017029933, 0,
1.01922074306095030425017029933, 2.22979463158591224986333623576, 3.52937480511297748516161487033, 4.65054718541626820691555921517, 5.12758042892308670012906404035, 5.62712514235505167104245614054, 6.46801161469179809986960092512, 7.32427127210245724502068790776, 7.59071503795007360547261420806
