| L(s) = 1 | + 42.5·2-s − 206.·3-s + 1.30e3·4-s − 1.78e3·5-s − 8.77e3·6-s − 1.16e4·7-s + 3.36e4·8-s + 2.27e4·9-s − 7.58e4·10-s + 1.44e4·11-s − 2.68e5·12-s − 2.67e4·13-s − 4.96e5·14-s + 3.66e5·15-s + 7.64e5·16-s − 2.69e5·17-s + 9.69e5·18-s − 2.66e4·19-s − 2.31e6·20-s + 2.40e6·21-s + 6.16e5·22-s − 2.79e5·23-s − 6.92e6·24-s + 1.21e6·25-s − 1.14e6·26-s − 6.37e5·27-s − 1.51e7·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 1.46·3-s + 2.54·4-s − 1.27·5-s − 2.76·6-s − 1.83·7-s + 2.90·8-s + 1.15·9-s − 2.39·10-s + 0.297·11-s − 3.73·12-s − 0.260·13-s − 3.45·14-s + 1.87·15-s + 2.91·16-s − 0.782·17-s + 2.17·18-s − 0.0468·19-s − 3.23·20-s + 2.69·21-s + 0.560·22-s − 0.208·23-s − 4.26·24-s + 0.623·25-s − 0.489·26-s − 0.230·27-s − 4.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 2.79e5T \) |
| good | 2 | \( 1 - 42.5T + 512T^{2} \) |
| 3 | \( 1 + 206.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.78e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.16e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.67e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.69e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.66e4T + 3.22e11T^{2} \) |
| 29 | \( 1 - 4.27e4T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.74e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.75e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.11e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.24e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.52e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.47e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 4.98e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.33e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.27e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.33e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.91e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.03e8T + 7.60e17T^{2} \) |
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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
−15.42285875980961390707271201934, −13.45333087207656576139170074006, −12.27916053132835234598346903108, −11.84290810689360755632225143627, −10.52677555392765520894855151559, −6.94642512502772083637980677888, −6.17872485492719070162590457342, −4.60701980280803885621530720604, −3.33756633282684842345024010743, 0,
3.33756633282684842345024010743, 4.60701980280803885621530720604, 6.17872485492719070162590457342, 6.94642512502772083637980677888, 10.52677555392765520894855151559, 11.84290810689360755632225143627, 12.27916053132835234598346903108, 13.45333087207656576139170074006, 15.42285875980961390707271201934
