| L(s) = 1 | + 8.06·2-s − 9·3-s + 32.9·4-s − 25·5-s − 72.5·6-s − 68.7·7-s + 8.02·8-s + 81·9-s − 201.·10-s + 770.·11-s − 296.·12-s + 169·13-s − 554.·14-s + 225·15-s − 991.·16-s + 1.54e3·17-s + 653.·18-s − 687.·19-s − 824.·20-s + 618.·21-s + 6.21e3·22-s + 3.49e3·23-s − 72.2·24-s + 625·25-s + 1.36e3·26-s − 729·27-s − 2.26e3·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s − 0.577·3-s + 1.03·4-s − 0.447·5-s − 0.822·6-s − 0.530·7-s + 0.0443·8-s + 0.333·9-s − 0.637·10-s + 1.92·11-s − 0.595·12-s + 0.277·13-s − 0.755·14-s + 0.258·15-s − 0.967·16-s + 1.29·17-s + 0.475·18-s − 0.436·19-s − 0.461·20-s + 0.306·21-s + 2.73·22-s + 1.37·23-s − 0.0255·24-s + 0.200·25-s + 0.395·26-s − 0.192·27-s − 0.546·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.432029721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.432029721\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 8.06T + 32T^{2} \) |
| 7 | \( 1 + 68.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 770.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.54e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 687.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.92e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−12.03920867908228466768833701050, −11.12692355991658737364258838670, −9.793136212061683810274661601032, −8.612939894422866572284957430973, −6.83213057129163849686211739986, −6.35495718999978052650371216636, −5.09204803635724684526954478976, −4.04497553736140199419971129478, −3.16134773665292810478977276996, −1.02058206721589873333653111688,
1.02058206721589873333653111688, 3.16134773665292810478977276996, 4.04497553736140199419971129478, 5.09204803635724684526954478976, 6.35495718999978052650371216636, 6.83213057129163849686211739986, 8.612939894422866572284957430973, 9.793136212061683810274661601032, 11.12692355991658737364258838670, 12.03920867908228466768833701050
