| L(s) = 1 | + 4·2-s + 7.93·3-s + 16·4-s − 25·5-s + 31.7·6-s + 221.·7-s + 64·8-s − 180.·9-s − 100·10-s + 508.·11-s + 126.·12-s + 148.·13-s + 884.·14-s − 198.·15-s + 256·16-s − 1.62e3·17-s − 720.·18-s + 1.65e3·19-s − 400·20-s + 1.75e3·21-s + 2.03e3·22-s − 529·23-s + 507.·24-s + 625·25-s + 592.·26-s − 3.35e3·27-s + 3.53e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.508·3-s + 0.5·4-s − 0.447·5-s + 0.359·6-s + 1.70·7-s + 0.353·8-s − 0.741·9-s − 0.316·10-s + 1.26·11-s + 0.254·12-s + 0.242·13-s + 1.20·14-s − 0.227·15-s + 0.250·16-s − 1.36·17-s − 0.524·18-s + 1.05·19-s − 0.223·20-s + 0.868·21-s + 0.895·22-s − 0.208·23-s + 0.179·24-s + 0.200·25-s + 0.171·26-s − 0.885·27-s + 0.853·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.397928750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.397928750\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 - 7.93T + 243T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 508.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 148.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.65e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 813.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.20e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.72e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.91e4T + 8.58e9T^{2} \) |
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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
−11.60071506600563680744775225441, −10.78715871526769016469161524768, −9.091168768412520960479839894609, −8.344945721753723393249068369878, −7.40136069179618910964275045362, −6.12084670260192579875059010386, −4.81350346114036403866319085927, −3.99011976966611501959124677601, −2.57271126645468812041509919043, −1.24616637375707119729299325130,
1.24616637375707119729299325130, 2.57271126645468812041509919043, 3.99011976966611501959124677601, 4.81350346114036403866319085927, 6.12084670260192579875059010386, 7.40136069179618910964275045362, 8.344945721753723393249068369878, 9.091168768412520960479839894609, 10.78715871526769016469161524768, 11.60071506600563680744775225441
