| L(s) = 1 | + 4·2-s − 22.9·3-s + 16·4-s − 25·5-s − 91.6·6-s + 10.2·7-s + 64·8-s + 281.·9-s − 100·10-s + 66.7·11-s − 366.·12-s − 1.13e3·13-s + 41.0·14-s + 572.·15-s + 256·16-s + 115.·17-s + 1.12e3·18-s − 1.87e3·19-s − 400·20-s − 235.·21-s + 266.·22-s − 529·23-s − 1.46e3·24-s + 625·25-s − 4.55e3·26-s − 885.·27-s + 164.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.447·5-s − 1.03·6-s + 0.0791·7-s + 0.353·8-s + 1.15·9-s − 0.316·10-s + 0.166·11-s − 0.734·12-s − 1.86·13-s + 0.0559·14-s + 0.657·15-s + 0.250·16-s + 0.0968·17-s + 0.819·18-s − 1.19·19-s − 0.223·20-s − 0.116·21-s + 0.117·22-s − 0.208·23-s − 0.519·24-s + 0.200·25-s − 1.32·26-s − 0.233·27-s + 0.0395·28-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
| \(L(3)\) |
\(\approx\) |
\(1.288094946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288094946\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 22.9T + 243T^{2} \) |
| 7 | \( 1 - 10.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 66.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 115.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.87e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 5.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.59e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.99e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.78e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−11.58339690021759432289755961077, −10.66654110696790477701049758193, −9.805564468021751936766872589227, −8.067913006832085178835001671061, −6.92418203304290334761178203335, −6.14366902623775062210920222663, −4.95015923898109777212434879538, −4.35836402128558288440068279683, −2.52243626416122265160113078457, −0.63897860712171483711501349852,
0.63897860712171483711501349852, 2.52243626416122265160113078457, 4.35836402128558288440068279683, 4.95015923898109777212434879538, 6.14366902623775062210920222663, 6.92418203304290334761178203335, 8.067913006832085178835001671061, 9.805564468021751936766872589227, 10.66654110696790477701049758193, 11.58339690021759432289755961077
