| L(s) = 1 | + 14.8·2-s + 536.·3-s − 1.82e3·4-s + 2.11e3·5-s + 7.98e3·6-s − 6.85e4·7-s − 5.76e4·8-s + 1.10e5·9-s + 3.15e4·10-s − 5.73e5·11-s − 9.80e5·12-s − 2.09e5·13-s − 1.01e6·14-s + 1.13e6·15-s + 2.88e6·16-s + 7.02e6·17-s + 1.65e6·18-s − 1.37e7·19-s − 3.87e6·20-s − 3.68e7·21-s − 8.53e6·22-s + 6.43e6·23-s − 3.09e7·24-s − 4.43e7·25-s − 3.11e6·26-s − 3.55e7·27-s + 1.25e8·28-s + ⋯ |
| L(s) = 1 | + 0.328·2-s + 1.27·3-s − 0.892·4-s + 0.303·5-s + 0.419·6-s − 1.54·7-s − 0.621·8-s + 0.626·9-s + 0.0996·10-s − 1.07·11-s − 1.13·12-s − 0.156·13-s − 0.506·14-s + 0.386·15-s + 0.687·16-s + 1.20·17-s + 0.205·18-s − 1.27·19-s − 0.270·20-s − 1.96·21-s − 0.353·22-s + 0.208·23-s − 0.792·24-s − 0.908·25-s − 0.0514·26-s − 0.476·27-s + 1.37·28-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
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 14.8T + 2.04e3T^{2} \) |
| 3 | \( 1 - 536.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 2.11e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.85e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.73e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.09e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.02e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.37e7T + 1.16e14T^{2} \) |
| 29 | \( 1 + 1.90e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.11e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.70e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.78e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.32e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.04e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.65e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.91e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.85e6T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.92e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 5.27e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.46e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 9.77e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.67e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.63e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 5.73e10T + 7.15e21T^{2} \) |
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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
−14.41191642318111039505868584990, −13.29667634265567325046459062498, −12.75026194717138107968125051477, −10.02863036371426798806421978660, −9.221824911806597330853921418229, −7.87490759469249487475653915701, −5.80117648738135086818556360645, −3.77909079558915768911204849044, −2.66577372529939635056589328317, 0,
2.66577372529939635056589328317, 3.77909079558915768911204849044, 5.80117648738135086818556360645, 7.87490759469249487475653915701, 9.221824911806597330853921418229, 10.02863036371426798806421978660, 12.75026194717138107968125051477, 13.29667634265567325046459062498, 14.41191642318111039505868584990
