| L(s) = 1 | + 1.70·2-s − 5.10·4-s + 5·5-s + 7·7-s − 22.2·8-s + 8.50·10-s − 37.4·11-s + 29.0·13-s + 11.9·14-s + 2.89·16-s − 58.4·17-s − 54.5·19-s − 25.5·20-s − 63.6·22-s − 161.·23-s + 25·25-s + 49.3·26-s − 35.7·28-s − 137.·29-s + 154.·31-s + 183.·32-s − 99.4·34-s + 35·35-s − 350.·37-s − 92.8·38-s − 111.·40-s − 353.·41-s + ⋯ |
| L(s) = 1 | + 0.601·2-s − 0.638·4-s + 0.447·5-s + 0.377·7-s − 0.985·8-s + 0.269·10-s − 1.02·11-s + 0.619·13-s + 0.227·14-s + 0.0452·16-s − 0.833·17-s − 0.659·19-s − 0.285·20-s − 0.616·22-s − 1.46·23-s + 0.200·25-s + 0.372·26-s − 0.241·28-s − 0.880·29-s + 0.896·31-s + 1.01·32-s − 0.501·34-s + 0.169·35-s − 1.55·37-s − 0.396·38-s − 0.440·40-s − 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.70T + 8T^{2} \) |
| 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
| 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
−10.71628834787991071714588334763, −9.915843558118206884024008710122, −8.749022061685460881585497054662, −8.112804230021044235271879989329, −6.57278425894894481640026427981, −5.57744502018219020086240826481, −4.71277872264457071646378309301, −3.57332554106405253211400630797, −2.06831059306442767213738978273, 0,
2.06831059306442767213738978273, 3.57332554106405253211400630797, 4.71277872264457071646378309301, 5.57744502018219020086240826481, 6.57278425894894481640026427981, 8.112804230021044235271879989329, 8.749022061685460881585497054662, 9.915843558118206884024008710122, 10.71628834787991071714588334763
