| L(s) = 1 | + 43.8·2-s + 468.·3-s + 898.·4-s + 2.05e4·6-s − 5.51e3·8-s + 1.60e5·9-s + 4.20e5·12-s − 4.39e5·13-s − 1.16e6·16-s + 7.03e6·18-s − 6.43e6·23-s − 2.58e6·24-s + 9.76e6·25-s − 1.92e7·26-s + 4.74e7·27-s + 3.35e7·29-s − 4.11e7·31-s − 4.52e7·32-s + 1.44e8·36-s − 2.05e8·39-s − 1.13e8·41-s − 2.82e8·46-s − 4.44e8·47-s − 5.44e8·48-s + 2.82e8·49-s + 4.28e8·50-s − 3.94e8·52-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 1.92·3-s + 0.877·4-s + 2.64·6-s − 0.168·8-s + 2.71·9-s + 1.69·12-s − 1.18·13-s − 1.10·16-s + 3.72·18-s − 23-s − 0.324·24-s + 25-s − 1.61·26-s + 3.30·27-s + 1.63·29-s − 1.43·31-s − 1.34·32-s + 2.38·36-s − 2.27·39-s − 0.977·41-s − 1.37·46-s − 1.93·47-s − 2.13·48-s + 49-s + 1.37·50-s − 1.03·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(6.163957729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.163957729\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 6.43e6T \) |
| good | 2 | \( 1 - 43.8T + 1.02e3T^{2} \) |
| 3 | \( 1 - 468.T + 5.90e4T^{2} \) |
| 5 | \( 1 - 9.76e6T^{2} \) |
| 7 | \( 1 - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.59e10T^{2} \) |
| 13 | \( 1 + 4.39e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.01e12T^{2} \) |
| 19 | \( 1 - 6.13e12T^{2} \) |
| 29 | \( 1 - 3.35e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 4.11e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.13e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.16e16T^{2} \) |
| 47 | \( 1 + 4.44e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 1.74e17T^{2} \) |
| 59 | \( 1 - 1.28e9T + 5.11e17T^{2} \) |
| 61 | \( 1 - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.94e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + 3.77e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 - 1.55e19T^{2} \) |
| 89 | \( 1 - 3.11e19T^{2} \) |
| 97 | \( 1 - 7.37e19T^{2} \) |
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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
−14.80663298689163643002375323164, −14.32218169430683482456300151817, −13.25585574177065497424156947403, −12.28493243983268572150318425400, −9.903201068103580495880253587665, −8.544381725300527933140158074582, −7.02153988491689090598419583352, −4.70509831346470153702793457898, −3.39471859988848083048221807809, −2.25549421085970956938734558925,
2.25549421085970956938734558925, 3.39471859988848083048221807809, 4.70509831346470153702793457898, 7.02153988491689090598419583352, 8.544381725300527933140158074582, 9.903201068103580495880253587665, 12.28493243983268572150318425400, 13.25585574177065497424156947403, 14.32218169430683482456300151817, 14.80663298689163643002375323164
