| L(s) = 1 | + 263.·3-s + 2.08e3·5-s + 3.94e3·7-s + 4.99e4·9-s − 3.27e4·11-s + 2.85e4·13-s + 5.49e5·15-s + 5.79e5·17-s − 9.09e5·19-s + 1.04e6·21-s − 1.86e6·23-s + 2.38e6·25-s + 7.98e6·27-s + 4.60e6·29-s + 2.51e6·31-s − 8.64e6·33-s + 8.23e6·35-s − 1.29e7·37-s + 7.53e6·39-s − 3.17e7·41-s − 2.64e7·43-s + 1.04e8·45-s − 2.69e7·47-s − 2.47e7·49-s + 1.52e8·51-s + 1.30e7·53-s − 6.82e7·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 1.49·5-s + 0.621·7-s + 2.53·9-s − 0.674·11-s + 0.277·13-s + 2.80·15-s + 1.68·17-s − 1.60·19-s + 1.16·21-s − 1.39·23-s + 1.22·25-s + 2.89·27-s + 1.20·29-s + 0.489·31-s − 1.26·33-s + 0.927·35-s − 1.13·37-s + 0.521·39-s − 1.75·41-s − 1.18·43-s + 3.78·45-s − 0.805·47-s − 0.613·49-s + 3.16·51-s + 0.226·53-s − 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.029973623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.029973623\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 263.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.08e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.27e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 5.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.86e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.17e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.64e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.91e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.36e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.56e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.79e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.94e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.96e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.74e7T + 7.60e17T^{2} \) |
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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
−12.40666669861969468691991287444, −10.20699134654312287030737013800, −9.971861015231742545539328828103, −8.541797611912972559921214591417, −8.059082312176583752572570914258, −6.50609234520481501488397721401, −4.97511565536809494250011883834, −3.42130435858223178660950400608, −2.21863361568293173988648424579, −1.55786037716839599100185494932,
1.55786037716839599100185494932, 2.21863361568293173988648424579, 3.42130435858223178660950400608, 4.97511565536809494250011883834, 6.50609234520481501488397721401, 8.059082312176583752572570914258, 8.541797611912972559921214591417, 9.971861015231742545539328828103, 10.20699134654312287030737013800, 12.40666669861969468691991287444
