| L(s) = 1 | + 0.109·2-s + 6.64·3-s − 7.98·4-s − 5·5-s + 0.729·6-s + 28.5·7-s − 1.75·8-s + 17.2·9-s − 0.548·10-s + 11·11-s − 53.1·12-s + 18.1·13-s + 3.13·14-s − 33.2·15-s + 63.7·16-s + 0.876·17-s + 1.88·18-s + 19·19-s + 39.9·20-s + 189.·21-s + 1.20·22-s − 88.8·23-s − 11.6·24-s + 25·25-s + 1.99·26-s − 65.1·27-s − 227.·28-s + ⋯ |
| L(s) = 1 | + 0.0388·2-s + 1.27·3-s − 0.998·4-s − 0.447·5-s + 0.0496·6-s + 1.54·7-s − 0.0775·8-s + 0.637·9-s − 0.0173·10-s + 0.301·11-s − 1.27·12-s + 0.387·13-s + 0.0598·14-s − 0.572·15-s + 0.995·16-s + 0.0125·17-s + 0.0247·18-s + 0.229·19-s + 0.446·20-s + 1.97·21-s + 0.0117·22-s − 0.805·23-s − 0.0992·24-s + 0.200·25-s + 0.0150·26-s − 0.464·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.111877563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.111877563\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 0.109T + 8T^{2} \) |
| 3 | \( 1 - 6.64T + 27T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 13 | \( 1 - 18.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.876T + 4.91e3T^{2} \) |
| 23 | \( 1 + 88.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 178.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 202.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 31.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 566.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 216.T + 9.12e5T^{2} \) |
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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
−9.205839871003798503591028525456, −8.681502271380248995188253429533, −7.929421753938054506120171597651, −7.66437846166524814895405816413, −6.02927862300512823227031922787, −4.87131398595025464191581304631, −4.20141449466330109517043488829, −3.38045622714646545007427735060, −2.10732258960970528614674907442, −0.926038327152865601721634879396,
0.926038327152865601721634879396, 2.10732258960970528614674907442, 3.38045622714646545007427735060, 4.20141449466330109517043488829, 4.87131398595025464191581304631, 6.02927862300512823227031922787, 7.66437846166524814895405816413, 7.929421753938054506120171597651, 8.681502271380248995188253429533, 9.205839871003798503591028525456
