| L(s) = 1 | + 10.8·2-s − 1.98·3-s + 86.5·4-s − 25·5-s − 21.6·6-s + 73.9·7-s + 593.·8-s − 239.·9-s − 272.·10-s + 121·11-s − 171.·12-s + 816.·13-s + 804.·14-s + 49.6·15-s + 3.69e3·16-s − 122.·17-s − 2.60e3·18-s − 361·19-s − 2.16e3·20-s − 146.·21-s + 1.31e3·22-s + 895.·23-s − 1.17e3·24-s + 625·25-s + 8.88e3·26-s + 957.·27-s + 6.39e3·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.127·3-s + 2.70·4-s − 0.447·5-s − 0.245·6-s + 0.570·7-s + 3.27·8-s − 0.983·9-s − 0.860·10-s + 0.301·11-s − 0.344·12-s + 1.33·13-s + 1.09·14-s + 0.0569·15-s + 3.60·16-s − 0.102·17-s − 1.89·18-s − 0.229·19-s − 1.20·20-s − 0.0726·21-s + 0.580·22-s + 0.353·23-s − 0.417·24-s + 0.200·25-s + 2.57·26-s + 0.252·27-s + 1.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.152040679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.152040679\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 10.8T + 32T^{2} \) |
| 3 | \( 1 + 1.98T + 243T^{2} \) |
| 7 | \( 1 - 73.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 816.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 122.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 895.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.74e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.16e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.48e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 397.T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.31e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.59e4T + 8.58e9T^{2} \) |
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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
−9.025276959832226156911896654748, −8.080973842252070421038093443615, −7.24663410300769415336374851371, −6.19352852948328876284399133991, −5.76650110994058963052598460142, −4.76471049984506683346951334267, −3.99419926707534508175072408775, −3.22177635456945920427280563641, −2.23090673453354784939670686253, −1.02665627229847561530835266241,
1.02665627229847561530835266241, 2.23090673453354784939670686253, 3.22177635456945920427280563641, 3.99419926707534508175072408775, 4.76471049984506683346951334267, 5.76650110994058963052598460142, 6.19352852948328876284399133991, 7.24663410300769415336374851371, 8.080973842252070421038093443615, 9.025276959832226156911896654748
