| L(s) = 1 | − 39.1·2-s − 17.2·3-s + 1.01e3·4-s + 2.45e3·5-s + 676.·6-s + 9.79e3·7-s − 1.97e4·8-s − 1.93e4·9-s − 9.58e4·10-s + 6.62e4·11-s − 1.75e4·12-s − 9.57e3·13-s − 3.83e5·14-s − 4.23e4·15-s + 2.51e5·16-s − 5.62e5·17-s + 7.58e5·18-s − 7.10e5·19-s + 2.49e6·20-s − 1.69e5·21-s − 2.59e6·22-s − 2.17e6·23-s + 3.41e5·24-s + 4.05e6·25-s + 3.74e5·26-s + 6.75e5·27-s + 9.96e6·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.123·3-s + 1.98·4-s + 1.75·5-s + 0.212·6-s + 1.54·7-s − 1.70·8-s − 0.984·9-s − 3.03·10-s + 1.36·11-s − 0.244·12-s − 0.0930·13-s − 2.66·14-s − 0.216·15-s + 0.961·16-s − 1.63·17-s + 1.70·18-s − 1.24·19-s + 3.48·20-s − 0.190·21-s − 2.35·22-s − 1.62·23-s + 0.210·24-s + 2.07·25-s + 0.160·26-s + 0.244·27-s + 3.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.496112298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496112298\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 - 1.98e9T \) |
| good | 2 | \( 1 + 39.1T + 512T^{2} \) |
| 3 | \( 1 + 17.2T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.45e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.79e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.62e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.57e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.62e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.10e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.08e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.08e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.78e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.97e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.14e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.87e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.32e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.25e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.15e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.18e9T + 7.60e17T^{2} \) |
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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
−10.60317151867580071955972111201, −9.579411759802948840606170773331, −8.718693627340651644277409964575, −8.312902276916400774002185784873, −6.64121417300395214331737248611, −6.12764045184754393413121845241, −4.65857063708480754693952836891, −2.13981342135036214650316530404, −1.98071785201949608534044286751, −0.76580114957128672517428908450,
0.76580114957128672517428908450, 1.98071785201949608534044286751, 2.13981342135036214650316530404, 4.65857063708480754693952836891, 6.12764045184754393413121845241, 6.64121417300395214331737248611, 8.312902276916400774002185784873, 8.718693627340651644277409964575, 9.579411759802948840606170773331, 10.60317151867580071955972111201
