L(s) = 1 | + 4.46·2-s + 9.80·3-s + 11.9·4-s − 5·5-s + 43.8·6-s + 17.6·8-s + 69.2·9-s − 22.3·10-s − 56.5·11-s + 117.·12-s + 40.9·13-s − 49.0·15-s − 16.6·16-s + 2.18·17-s + 309.·18-s + 16.4·19-s − 59.7·20-s − 252.·22-s − 155.·23-s + 173.·24-s + 25·25-s + 183.·26-s + 414.·27-s − 6.26·29-s − 219.·30-s + 168.·31-s − 215.·32-s + ⋯ |
L(s) = 1 | + 1.57·2-s + 1.88·3-s + 1.49·4-s − 0.447·5-s + 2.98·6-s + 0.781·8-s + 2.56·9-s − 0.706·10-s − 1.54·11-s + 2.82·12-s + 0.873·13-s − 0.844·15-s − 0.260·16-s + 0.0312·17-s + 4.04·18-s + 0.198·19-s − 0.668·20-s − 2.44·22-s − 1.40·23-s + 1.47·24-s + 0.200·25-s + 1.38·26-s + 2.95·27-s − 0.0400·29-s − 1.33·30-s + 0.977·31-s − 1.19·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(6.786901902\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.786901902\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 4.46T + 8T^{2} \) |
| 3 | \( 1 - 9.80T + 27T^{2} \) |
| 11 | \( 1 + 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.26T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 934.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 752.T + 9.12e5T^{2} \) |
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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
−12.16411677403737909495104306173, −10.80055542375259916580304044283, −9.711465170946422691362550975979, −8.353364261499241539803393341360, −7.82765693874826438970777311765, −6.56411547477754575625049557055, −5.04380128049639869261488655691, −3.93172804780944131526303425469, −3.16501718602807753075128091919, −2.16401220417760296020271095160,
2.16401220417760296020271095160, 3.16501718602807753075128091919, 3.93172804780944131526303425469, 5.04380128049639869261488655691, 6.56411547477754575625049557055, 7.82765693874826438970777311765, 8.353364261499241539803393341360, 9.711465170946422691362550975979, 10.80055542375259916580304044283, 12.16411677403737909495104306173