| L(s) = 1 | + 4·2-s + 16·4-s + 79.4·5-s + 132.·7-s + 64·8-s + 317.·10-s − 311.·11-s + 901.·13-s + 531.·14-s + 256·16-s + 157.·17-s + 361·19-s + 1.27e3·20-s − 1.24e3·22-s + 2.52e3·23-s + 3.18e3·25-s + 3.60e3·26-s + 2.12e3·28-s − 4.73e3·29-s − 6.58e3·31-s + 1.02e3·32-s + 631.·34-s + 1.05e4·35-s + 8.50e3·37-s + 1.44e3·38-s + 5.08e3·40-s − 1.97e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.42·5-s + 1.02·7-s + 0.353·8-s + 1.00·10-s − 0.776·11-s + 1.47·13-s + 0.724·14-s + 0.250·16-s + 0.132·17-s + 0.229·19-s + 0.710·20-s − 0.548·22-s + 0.994·23-s + 1.01·25-s + 1.04·26-s + 0.512·28-s − 1.04·29-s − 1.23·31-s + 0.176·32-s + 0.0936·34-s + 1.45·35-s + 1.02·37-s + 0.162·38-s + 0.502·40-s − 1.83·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.243270891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.243270891\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 5 | \( 1 - 79.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 132.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 311.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 157.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.97e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.15e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.39e5T + 8.58e9T^{2} \) |
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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
−10.94072493654712365138999879599, −9.902718518353493938872666217647, −8.834421772874892870690543963407, −7.80785673508804287938946889734, −6.56749796627068236251096466261, −5.58125101861427211173378995695, −5.03574637699690765834961085774, −3.53696331820811721479857001388, −2.17715654927362662653775419862, −1.30266828917929446264437504081,
1.30266828917929446264437504081, 2.17715654927362662653775419862, 3.53696331820811721479857001388, 5.03574637699690765834961085774, 5.58125101861427211173378995695, 6.56749796627068236251096466261, 7.80785673508804287938946889734, 8.834421772874892870690543963407, 9.902718518353493938872666217647, 10.94072493654712365138999879599
