| L(s) = 1 | + 16.8·2-s − 226.·4-s + 2.43e3·5-s − 2.40e3·7-s − 1.24e4·8-s + 4.11e4·10-s + 2.85e4·11-s + 1.38e5·13-s − 4.05e4·14-s − 9.47e4·16-s + 1.01e5·17-s − 4.88e5·19-s − 5.52e5·20-s + 4.82e5·22-s + 1.40e5·23-s + 3.99e6·25-s + 2.33e6·26-s + 5.44e5·28-s + 6.31e6·29-s − 1.00e6·31-s + 4.78e6·32-s + 1.70e6·34-s − 5.85e6·35-s + 1.19e7·37-s − 8.25e6·38-s − 3.04e7·40-s + 2.15e7·41-s + ⋯ |
| L(s) = 1 | + 0.746·2-s − 0.442·4-s + 1.74·5-s − 0.377·7-s − 1.07·8-s + 1.30·10-s + 0.587·11-s + 1.34·13-s − 0.282·14-s − 0.361·16-s + 0.293·17-s − 0.860·19-s − 0.772·20-s + 0.438·22-s + 0.104·23-s + 2.04·25-s + 1.00·26-s + 0.167·28-s + 1.65·29-s − 0.196·31-s + 0.807·32-s + 0.218·34-s − 0.659·35-s + 1.04·37-s − 0.642·38-s − 1.87·40-s + 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.514155982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.514155982\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 16.8T + 512T^{2} \) |
| 5 | \( 1 - 2.43e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 2.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.38e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.01e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.40e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.19e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.65e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.81e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.50e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.18e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.89e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.68e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.75e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.36e8T + 7.60e17T^{2} \) |
| 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
−13.30395682107059227481721473065, −12.38763670566049960785368817548, −10.68329679775750014404255691734, −9.515105265928074291162687888730, −8.723354408451158818185602100528, −6.34317591265805944725595352958, −5.83420409501913154825657017499, −4.32609879453086827627036105595, −2.79461937704234022375712843980, −1.14536603437960030537564022318,
1.14536603437960030537564022318, 2.79461937704234022375712843980, 4.32609879453086827627036105595, 5.83420409501913154825657017499, 6.34317591265805944725595352958, 8.723354408451158818185602100528, 9.515105265928074291162687888730, 10.68329679775750014404255691734, 12.38763670566049960785368817548, 13.30395682107059227481721473065
