| L(s) = 1 | − 4·2-s − 7.77·3-s + 16·4-s + 31.1·6-s − 49·7-s − 64·8-s − 182.·9-s − 588.·11-s − 124.·12-s − 147.·13-s + 196·14-s + 256·16-s − 63.1·17-s + 730.·18-s − 1.61e3·19-s + 381.·21-s + 2.35e3·22-s + 1.48e3·23-s + 497.·24-s + 590.·26-s + 3.30e3·27-s − 784·28-s − 1.69e3·29-s − 7.44e3·31-s − 1.02e3·32-s + 4.57e3·33-s + 252.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.498·3-s + 0.5·4-s + 0.352·6-s − 0.377·7-s − 0.353·8-s − 0.751·9-s − 1.46·11-s − 0.249·12-s − 0.242·13-s + 0.267·14-s + 0.250·16-s − 0.0530·17-s + 0.531·18-s − 1.02·19-s + 0.188·21-s + 1.03·22-s + 0.585·23-s + 0.176·24-s + 0.171·26-s + 0.873·27-s − 0.188·28-s − 0.373·29-s − 1.39·31-s − 0.176·32-s + 0.731·33-s + 0.0374·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3397311837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3397311837\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 7.77T + 243T^{2} \) |
| 11 | \( 1 + 588.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 63.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 334.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.86e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.86e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.06e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.07e4T + 8.58e9T^{2} \) |
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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
−10.77660990074949290357877825893, −9.797621027459837193170485978171, −8.776625447618634591824818280420, −7.938149967772523902785291675540, −6.90064993649223603580672891589, −5.86432042534986802348517008795, −4.97687701048653956017438682014, −3.23069299962421065946795879124, −2.12672100287617238692924190592, −0.33790409625047817465841841411,
0.33790409625047817465841841411, 2.12672100287617238692924190592, 3.23069299962421065946795879124, 4.97687701048653956017438682014, 5.86432042534986802348517008795, 6.90064993649223603580672891589, 7.938149967772523902785291675540, 8.776625447618634591824818280420, 9.797621027459837193170485978171, 10.77660990074949290357877825893
