| L(s) = 1 | + 3.53·2-s − 19.5·4-s + 25·5-s − 49·7-s − 181.·8-s + 88.2·10-s + 691.·11-s − 502.·13-s − 173.·14-s − 17.5·16-s + 991.·17-s + 661.·19-s − 488.·20-s + 2.44e3·22-s − 3.41e3·23-s + 625·25-s − 1.77e3·26-s + 957.·28-s − 6.75e3·29-s − 3.92e3·31-s + 5.76e3·32-s + 3.50e3·34-s − 1.22e3·35-s + 627.·37-s + 2.33e3·38-s − 4.54e3·40-s − 1.62e4·41-s + ⋯ |
| L(s) = 1 | + 0.624·2-s − 0.610·4-s + 0.447·5-s − 0.377·7-s − 1.00·8-s + 0.279·10-s + 1.72·11-s − 0.824·13-s − 0.235·14-s − 0.0171·16-s + 0.831·17-s + 0.420·19-s − 0.272·20-s + 1.07·22-s − 1.34·23-s + 0.200·25-s − 0.514·26-s + 0.230·28-s − 1.49·29-s − 0.733·31-s + 0.994·32-s + 0.519·34-s − 0.169·35-s + 0.0753·37-s + 0.262·38-s − 0.449·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 2 | \( 1 - 3.53T + 32T^{2} \) |
| 11 | \( 1 - 691.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 502.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 991.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 661.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 627.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.25e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04913785178193353363361755884, −9.520440612253609428201484253248, −8.677638801117669153065151842697, −7.28979031650323968124547964654, −6.16849123800573892621289867332, −5.36575485034628472963006233604, −4.13135456458158458090289048099, −3.29832405754371658664629042988, −1.61463199393031977078664904262, 0,
1.61463199393031977078664904262, 3.29832405754371658664629042988, 4.13135456458158458090289048099, 5.36575485034628472963006233604, 6.16849123800573892621289867332, 7.28979031650323968124547964654, 8.677638801117669153065151842697, 9.520440612253609428201484253248, 10.04913785178193353363361755884
