| L(s) = 1 | − 3.40·2-s + 746.·3-s − 2.03e3·4-s − 1.14e4·5-s − 2.54e3·6-s + 3.86e4·7-s + 1.39e4·8-s + 3.80e5·9-s + 3.89e4·10-s + 7.33e5·11-s − 1.52e6·12-s + 1.73e6·13-s − 1.31e5·14-s − 8.53e6·15-s + 4.12e6·16-s − 7.09e5·17-s − 1.29e6·18-s + 4.74e6·19-s + 2.32e7·20-s + 2.88e7·21-s − 2.49e6·22-s − 6.43e6·23-s + 1.03e7·24-s + 8.17e7·25-s − 5.91e6·26-s + 1.51e8·27-s − 7.86e7·28-s + ⋯ |
| L(s) = 1 | − 0.0752·2-s + 1.77·3-s − 0.994·4-s − 1.63·5-s − 0.133·6-s + 0.868·7-s + 0.150·8-s + 2.14·9-s + 0.123·10-s + 1.37·11-s − 1.76·12-s + 1.29·13-s − 0.0653·14-s − 2.90·15-s + 0.983·16-s − 0.121·17-s − 0.161·18-s + 0.439·19-s + 1.62·20-s + 1.54·21-s − 0.103·22-s − 0.208·23-s + 0.266·24-s + 1.67·25-s − 0.0976·26-s + 2.03·27-s − 0.864·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.513447564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.513447564\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 6.43e6T \) |
| good | 2 | \( 1 + 3.40T + 2.04e3T^{2} \) |
| 3 | \( 1 - 746.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.14e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.86e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.33e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.73e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.09e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.74e6T + 1.16e14T^{2} \) |
| 29 | \( 1 + 4.78e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.42e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.88e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.50e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.52e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.47e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 8.04e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 8.78e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.87e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.69e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.27e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.74e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.59e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.05e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.36e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.61e10T + 7.15e21T^{2} \) |
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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
−14.80724496120564602266005953317, −14.30623822506695264318823184666, −12.96500994942544508243876647696, −11.35350266652572856381414591796, −9.220919584060823312081363450276, −8.429843159719380179210466863241, −7.58533352788874773867267821781, −4.18003504441529912615311393460, −3.63669267560571451309218530662, −1.20727324447892347666792247959,
1.20727324447892347666792247959, 3.63669267560571451309218530662, 4.18003504441529912615311393460, 7.58533352788874773867267821781, 8.429843159719380179210466863241, 9.220919584060823312081363450276, 11.35350266652572856381414591796, 12.96500994942544508243876647696, 14.30623822506695264318823184666, 14.80724496120564602266005953317
