| L(s) = 1 | + 2.12·2-s − 3.47·4-s + 5·5-s + 30.7·7-s − 24.4·8-s + 10.6·10-s + 50.1·11-s − 15.9·13-s + 65.2·14-s − 24.0·16-s + 105.·17-s − 21.3·19-s − 17.3·20-s + 106.·22-s + 136.·23-s + 25·25-s − 33.9·26-s − 106.·28-s − 224.·29-s − 225.·31-s + 144.·32-s + 224.·34-s + 153.·35-s − 416.·37-s − 45.2·38-s − 122.·40-s + 76.1·41-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 0.434·4-s + 0.447·5-s + 1.65·7-s − 1.07·8-s + 0.336·10-s + 1.37·11-s − 0.340·13-s + 1.24·14-s − 0.375·16-s + 1.50·17-s − 0.257·19-s − 0.194·20-s + 1.03·22-s + 1.23·23-s + 0.200·25-s − 0.255·26-s − 0.720·28-s − 1.43·29-s − 1.30·31-s + 0.796·32-s + 1.13·34-s + 0.741·35-s − 1.85·37-s − 0.193·38-s − 0.482·40-s + 0.290·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.591288573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591288573\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 2.12T + 8T^{2} \) |
| 7 | \( 1 - 30.7T + 343T^{2} \) |
| 11 | \( 1 - 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 76.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 95.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 87.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 412.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 248.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 291.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 198.T + 9.12e5T^{2} \) |
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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
−12.77127949819881593737221918133, −11.91726111262064357755632527050, −10.96913094984489451978254615327, −9.486032794316793730076526843229, −8.654723645012659809167143009417, −7.30648946861892812463628468798, −5.69352480564299846496836785901, −4.88656520652513150134616365760, −3.61926941319441371162782051718, −1.51615447186264782629634075631,
1.51615447186264782629634075631, 3.61926941319441371162782051718, 4.88656520652513150134616365760, 5.69352480564299846496836785901, 7.30648946861892812463628468798, 8.654723645012659809167143009417, 9.486032794316793730076526843229, 10.96913094984489451978254615327, 11.91726111262064357755632527050, 12.77127949819881593737221918133
