| L(s) = 1 | − 4.07·2-s − 6.58·3-s − 15.3·4-s + 25·5-s + 26.8·6-s + 210.·7-s + 193.·8-s − 199.·9-s − 101.·10-s − 121·11-s + 101.·12-s + 824.·13-s − 856.·14-s − 164.·15-s − 296.·16-s − 522.·17-s + 814.·18-s − 361·19-s − 383.·20-s − 1.38e3·21-s + 493.·22-s + 2.47e3·23-s − 1.27e3·24-s + 625·25-s − 3.36e3·26-s + 2.91e3·27-s − 3.22e3·28-s + ⋯ |
| L(s) = 1 | − 0.721·2-s − 0.422·3-s − 0.479·4-s + 0.447·5-s + 0.304·6-s + 1.62·7-s + 1.06·8-s − 0.821·9-s − 0.322·10-s − 0.301·11-s + 0.202·12-s + 1.35·13-s − 1.16·14-s − 0.188·15-s − 0.289·16-s − 0.438·17-s + 0.592·18-s − 0.229·19-s − 0.214·20-s − 0.684·21-s + 0.217·22-s + 0.977·23-s − 0.450·24-s + 0.200·25-s − 0.975·26-s + 0.769·27-s − 0.777·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.658950004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658950004\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 4.07T + 32T^{2} \) |
| 3 | \( 1 + 6.58T + 243T^{2} \) |
| 7 | \( 1 - 210.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 824.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 522.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.88e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.44e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.96e4T + 8.58e9T^{2} \) |
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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
−8.881274827062543284407339376140, −8.501811929674547397010407180713, −7.916032861515663224053315084846, −6.69937768269664399960432499872, −5.65704861928148236431755086676, −4.95583115290597589963031855615, −4.16997578243807034648707262124, −2.60830920507504715845098271497, −1.38621589059102934636977146574, −0.73163877509741579414023393630,
0.73163877509741579414023393630, 1.38621589059102934636977146574, 2.60830920507504715845098271497, 4.16997578243807034648707262124, 4.95583115290597589963031855615, 5.65704861928148236431755086676, 6.69937768269664399960432499872, 7.916032861515663224053315084846, 8.501811929674547397010407180713, 8.881274827062543284407339376140
