| L(s) = 1 | − 3.21·2-s + 3·3-s + 2.35·4-s − 5·5-s − 9.65·6-s + 7·7-s + 18.1·8-s + 9·9-s + 16.0·10-s + 11.1·11-s + 7.07·12-s + 28.4·13-s − 22.5·14-s − 15·15-s − 77.3·16-s + 33.4·17-s − 28.9·18-s + 45.3·19-s − 11.7·20-s + 21·21-s − 36.0·22-s − 23·23-s + 54.4·24-s + 25·25-s − 91.5·26-s + 27·27-s + 16.5·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.577·3-s + 0.294·4-s − 0.447·5-s − 0.656·6-s + 0.377·7-s + 0.802·8-s + 0.333·9-s + 0.508·10-s + 0.306·11-s + 0.170·12-s + 0.606·13-s − 0.430·14-s − 0.258·15-s − 1.20·16-s + 0.476·17-s − 0.379·18-s + 0.547·19-s − 0.131·20-s + 0.218·21-s − 0.348·22-s − 0.208·23-s + 0.463·24-s + 0.200·25-s − 0.690·26-s + 0.192·27-s + 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.575099754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575099754\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 3.21T + 8T^{2} \) |
| 11 | \( 1 - 11.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 34.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 179.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 345.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 24.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 596.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 504.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 109.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 607.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.656526625318838146946921974842, −7.942196426494389655023615526651, −7.50172996608186608248826233473, −6.62389666993717986393046688327, −5.48797369711417574335074417269, −4.42938262873887561416409274945, −3.76040314675063742251867276316, −2.60423500357988285997859010277, −1.45321584694050101535534329117, −0.71478794089498166050320465071,
0.71478794089498166050320465071, 1.45321584694050101535534329117, 2.60423500357988285997859010277, 3.76040314675063742251867276316, 4.42938262873887561416409274945, 5.48797369711417574335074417269, 6.62389666993717986393046688327, 7.50172996608186608248826233473, 7.942196426494389655023615526651, 8.656526625318838146946921974842
