| L(s) = 1 | + 4.18·2-s − 14.4·4-s + 25·5-s + 86.0·7-s − 194.·8-s + 104.·10-s − 121·11-s + 47.8·13-s + 360.·14-s − 350.·16-s + 813.·17-s + 1.55e3·19-s − 362.·20-s − 506.·22-s − 4.11e3·23-s + 625·25-s + 200.·26-s − 1.24e3·28-s + 87.5·29-s − 6.32e3·31-s + 4.75e3·32-s + 3.40e3·34-s + 2.15e3·35-s + 1.56e4·37-s + 6.49e3·38-s − 4.86e3·40-s + 1.39e4·41-s + ⋯ |
| L(s) = 1 | + 0.739·2-s − 0.452·4-s + 0.447·5-s + 0.663·7-s − 1.07·8-s + 0.330·10-s − 0.301·11-s + 0.0784·13-s + 0.491·14-s − 0.342·16-s + 0.682·17-s + 0.986·19-s − 0.202·20-s − 0.223·22-s − 1.62·23-s + 0.200·25-s + 0.0580·26-s − 0.300·28-s + 0.0193·29-s − 1.18·31-s + 0.821·32-s + 0.505·34-s + 0.296·35-s + 1.87·37-s + 0.729·38-s − 0.480·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.952268981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952268981\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 4.18T + 32T^{2} \) |
| 7 | \( 1 - 86.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 47.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 813.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 87.5T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.56e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.83e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.52e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 245.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.99e4T + 8.58e9T^{2} \) |
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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
−10.02041331958515248112201315248, −9.388164602408630986599709553713, −8.304250636408006325729834659824, −7.49696506267796408500749470929, −6.01150578008553660247525438861, −5.46469167384275760252125517518, −4.47710552573339500536155101740, −3.49426425032971653958823125028, −2.23271834935140258623330523184, −0.789888518056409310696596779150,
0.789888518056409310696596779150, 2.23271834935140258623330523184, 3.49426425032971653958823125028, 4.47710552573339500536155101740, 5.46469167384275760252125517518, 6.01150578008553660247525438861, 7.49696506267796408500749470929, 8.304250636408006325729834659824, 9.388164602408630986599709553713, 10.02041331958515248112201315248
