| L(s) = 1 | + 4.94·2-s + 5.84·3-s − 7.50·4-s + 25·5-s + 28.9·6-s + 1.89·7-s − 195.·8-s − 208.·9-s + 123.·10-s − 43.8·12-s + 378.·13-s + 9.37·14-s + 146.·15-s − 727.·16-s + 1.08e3·17-s − 1.03e3·18-s + 3.11e3·19-s − 187.·20-s + 11.0·21-s − 3.60e3·23-s − 1.14e3·24-s + 625·25-s + 1.87e3·26-s − 2.63e3·27-s − 14.2·28-s − 2.83e3·29-s + 722.·30-s + ⋯ |
| L(s) = 1 | + 0.874·2-s + 0.374·3-s − 0.234·4-s + 0.447·5-s + 0.327·6-s + 0.0146·7-s − 1.08·8-s − 0.859·9-s + 0.391·10-s − 0.0878·12-s + 0.620·13-s + 0.0127·14-s + 0.167·15-s − 0.710·16-s + 0.909·17-s − 0.752·18-s + 1.98·19-s − 0.104·20-s + 0.00547·21-s − 1.42·23-s − 0.404·24-s + 0.200·25-s + 0.543·26-s − 0.696·27-s − 0.00342·28-s − 0.625·29-s + 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.94T + 32T^{2} \) |
| 3 | \( 1 - 5.84T + 243T^{2} \) |
| 7 | \( 1 - 1.89T + 1.68e4T^{2} \) |
| 13 | \( 1 - 378.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 3.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.87e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.59e4T + 8.58e9T^{2} \) |
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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
−9.515092245228959168498989211535, −8.535415810784822465233732958859, −7.80756798518145822835551972183, −6.36092911472789402033372822131, −5.63125433976170254018259590882, −4.91787477665401634141213497503, −3.50492391883348220076434058196, −3.07890875540125723186082000060, −1.54413723343761466597688320270, 0,
1.54413723343761466597688320270, 3.07890875540125723186082000060, 3.50492391883348220076434058196, 4.91787477665401634141213497503, 5.63125433976170254018259590882, 6.36092911472789402033372822131, 7.80756798518145822835551972183, 8.535415810784822465233732958859, 9.515092245228959168498989211535
