| L(s) = 1 | + 13.8·2-s + 63.2·4-s + 149.·5-s + 14.2·7-s − 895.·8-s + 2.06e3·10-s − 531.·11-s + 4.16e3·13-s + 197.·14-s − 2.04e4·16-s − 4.56e3·17-s + 3.18e4·19-s + 9.45e3·20-s − 7.35e3·22-s + 5.60e4·23-s − 5.57e4·25-s + 5.76e4·26-s + 903.·28-s − 5.20e4·29-s − 4.40e4·31-s − 1.68e5·32-s − 6.30e4·34-s + 2.13e3·35-s + 1.13e4·37-s + 4.40e5·38-s − 1.33e5·40-s + 4.55e5·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.494·4-s + 0.534·5-s + 0.0157·7-s − 0.618·8-s + 0.653·10-s − 0.120·11-s + 0.526·13-s + 0.0192·14-s − 1.24·16-s − 0.225·17-s + 1.06·19-s + 0.264·20-s − 0.147·22-s + 0.960·23-s − 0.714·25-s + 0.643·26-s + 0.00777·28-s − 0.396·29-s − 0.265·31-s − 0.909·32-s − 0.275·34-s + 0.00841·35-s + 0.0367·37-s + 1.30·38-s − 0.330·40-s + 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.615478543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.615478543\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 13.8T + 128T^{2} \) |
| 5 | \( 1 - 149.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 14.2T + 8.23e5T^{2} \) |
| 11 | \( 1 + 531.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.56e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.18e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.20e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.40e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.13e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.27e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.74e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.15e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.33e6T + 8.07e13T^{2} \) |
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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
−9.585513171032354412871939055441, −9.014401993268072625850112177047, −7.75145659856263171282948135033, −6.66220826971991114272067507340, −5.76975804775297893303718395190, −5.14910437375397385950225267169, −4.07392489979540425662420780603, −3.18484990313171452671914396152, −2.15126366664319611321598953264, −0.77412434481082378081434135360,
0.77412434481082378081434135360, 2.15126366664319611321598953264, 3.18484990313171452671914396152, 4.07392489979540425662420780603, 5.14910437375397385950225267169, 5.76975804775297893303718395190, 6.66220826971991114272067507340, 7.75145659856263171282948135033, 9.014401993268072625850112177047, 9.585513171032354412871939055441
