| L(s) = 1 | − 16·2-s + 256·4-s − 745.·5-s − 3.11e3·7-s − 4.09e3·8-s + 1.19e4·10-s + 7.10e3·11-s − 6.93e3·13-s + 4.98e4·14-s + 6.55e4·16-s + 3.27e5·17-s − 1.30e5·19-s − 1.90e5·20-s − 1.13e5·22-s + 9.55e5·23-s − 1.39e6·25-s + 1.10e5·26-s − 7.97e5·28-s − 2.95e6·29-s + 6.62e6·31-s − 1.04e6·32-s − 5.23e6·34-s + 2.32e6·35-s − 1.94e7·37-s + 2.08e6·38-s + 3.05e6·40-s − 2.62e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.533·5-s − 0.490·7-s − 0.353·8-s + 0.377·10-s + 0.146·11-s − 0.0673·13-s + 0.346·14-s + 0.250·16-s + 0.950·17-s − 0.229·19-s − 0.266·20-s − 0.103·22-s + 0.711·23-s − 0.715·25-s + 0.0476·26-s − 0.245·28-s − 0.776·29-s + 1.28·31-s − 0.176·32-s − 0.672·34-s + 0.261·35-s − 1.70·37-s + 0.162·38-s + 0.188·40-s − 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9293795799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9293795799\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 5 | \( 1 + 745.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.11e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.10e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.93e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.27e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 9.55e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.95e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.94e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.62e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.17e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.66e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.58e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.80e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.20e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.10e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.04e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.940739922084699582655826432596, −9.067124668620405756688402273554, −8.133899911440161517433945796910, −7.30625561855794580028435745648, −6.39216486101718159414163058690, −5.25537391466583375092251331663, −3.85702272346001500608119810833, −2.94240935072498381452255290086, −1.60759036619272813824762224592, −0.46722222011725045694362889736,
0.46722222011725045694362889736, 1.60759036619272813824762224592, 2.94240935072498381452255290086, 3.85702272346001500608119810833, 5.25537391466583375092251331663, 6.39216486101718159414163058690, 7.30625561855794580028435745648, 8.133899911440161517433945796910, 9.067124668620405756688402273554, 9.940739922084699582655826432596
