| L(s) = 1 | + 52.4·2-s + 243·3-s + 700.·4-s + 6.87e3·5-s + 1.27e4·6-s + 3.59e4·7-s − 7.06e4·8-s + 5.90e4·9-s + 3.60e5·10-s + 2.39e5·11-s + 1.70e5·12-s + 1.50e6·13-s + 1.88e6·14-s + 1.66e6·15-s − 5.13e6·16-s − 6.10e6·17-s + 3.09e6·18-s + 1.76e7·19-s + 4.81e6·20-s + 8.73e6·21-s + 1.25e7·22-s + 3.61e7·23-s − 1.71e7·24-s − 1.60e6·25-s + 7.88e7·26-s + 1.43e7·27-s + 2.51e7·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.577·3-s + 0.341·4-s + 0.983·5-s + 0.668·6-s + 0.808·7-s − 0.762·8-s + 0.333·9-s + 1.13·10-s + 0.447·11-s + 0.197·12-s + 1.12·13-s + 0.936·14-s + 0.567·15-s − 1.22·16-s − 1.04·17-s + 0.386·18-s + 1.63·19-s + 0.336·20-s + 0.466·21-s + 0.518·22-s + 1.17·23-s − 0.440·24-s − 0.0328·25-s + 1.30·26-s + 0.192·27-s + 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(7.418510051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.418510051\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 52.4T + 2.04e3T^{2} \) |
| 5 | \( 1 - 6.87e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.59e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.39e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.50e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.76e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.61e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.35e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.11e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.49e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.08e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.45e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.38e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.51e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 1.11e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.24e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.21e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.27e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.34e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.64e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.04e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.29e11T + 7.15e21T^{2} \) |
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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
−10.93311519651197650447151735694, −9.357852882499834253473997430598, −8.942707829816485098041497241753, −7.47381658940560527272041056150, −6.18346147270908680507199107453, −5.37406684290906411176423815910, −4.32092416794711810250500203301, −3.30764841280480524995277445383, −2.16260500984583392076723254886, −1.09669647993396878110403401943,
1.09669647993396878110403401943, 2.16260500984583392076723254886, 3.30764841280480524995277445383, 4.32092416794711810250500203301, 5.37406684290906411176423815910, 6.18346147270908680507199107453, 7.47381658940560527272041056150, 8.942707829816485098041497241753, 9.357852882499834253473997430598, 10.93311519651197650447151735694
