| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 62.6·5-s + 36·6-s − 69.5·7-s + 64·8-s + 81·9-s + 250.·10-s − 52.2·11-s + 144·12-s − 61.0·13-s − 278.·14-s + 563.·15-s + 256·16-s + 1.81e3·17-s + 324·18-s − 962.·19-s + 1.00e3·20-s − 625.·21-s − 209.·22-s + 3.39e3·23-s + 576·24-s + 798.·25-s − 244.·26-s + 729·27-s − 1.11e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.12·5-s + 0.408·6-s − 0.536·7-s + 0.353·8-s + 0.333·9-s + 0.792·10-s − 0.130·11-s + 0.288·12-s − 0.100·13-s − 0.379·14-s + 0.646·15-s + 0.250·16-s + 1.52·17-s + 0.235·18-s − 0.611·19-s + 0.560·20-s − 0.309·21-s − 0.0920·22-s + 1.33·23-s + 0.204·24-s + 0.255·25-s − 0.0708·26-s + 0.192·27-s − 0.268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.244924096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.244924096\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 - 62.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 69.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 52.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 61.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.81e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 962.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.02e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.38e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.94e4T + 8.58e9T^{2} \) |
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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
−10.39582297008500041136959616097, −9.883864172033255707753536342944, −8.884686734194487986239662338858, −7.70746418341878839940973146370, −6.60634964228571464590027191215, −5.78046561543293591153648287775, −4.72813282264095833399603755478, −3.32599523751118780418959064784, −2.49597086574008500791247233232, −1.19547704387149742117817820068,
1.19547704387149742117817820068, 2.49597086574008500791247233232, 3.32599523751118780418959064784, 4.72813282264095833399603755478, 5.78046561543293591153648287775, 6.60634964228571464590027191215, 7.70746418341878839940973146370, 8.884686734194487986239662338858, 9.883864172033255707753536342944, 10.39582297008500041136959616097
