| L(s) = 1 | − 6.60·2-s − 84.4·3-s − 84.3·4-s − 124.·5-s + 558.·6-s − 780.·7-s + 1.40e3·8-s + 4.94e3·9-s + 821.·10-s − 5.63e3·11-s + 7.11e3·12-s − 1.02e3·13-s + 5.16e3·14-s + 1.05e4·15-s + 1.51e3·16-s + 1.58e4·17-s − 3.26e4·18-s − 4.12e4·19-s + 1.04e4·20-s + 6.59e4·21-s + 3.72e4·22-s − 1.21e4·23-s − 1.18e5·24-s − 6.26e4·25-s + 6.74e3·26-s − 2.32e5·27-s + 6.58e4·28-s + ⋯ |
| L(s) = 1 | − 0.584·2-s − 1.80·3-s − 0.658·4-s − 0.444·5-s + 1.05·6-s − 0.860·7-s + 0.969·8-s + 2.26·9-s + 0.259·10-s − 1.27·11-s + 1.18·12-s − 0.128·13-s + 0.502·14-s + 0.803·15-s + 0.0925·16-s + 0.781·17-s − 1.32·18-s − 1.37·19-s + 0.293·20-s + 1.55·21-s + 0.745·22-s − 0.208·23-s − 1.74·24-s − 0.802·25-s + 0.0752·26-s − 2.27·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2152576625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2152576625\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 6.60T + 128T^{2} \) |
| 3 | \( 1 + 84.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 124.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 780.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.02e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.58e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.12e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.11e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.62e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.63e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.05e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.50e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.69e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.25e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.74e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.16e6T + 8.07e13T^{2} \) |
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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
−16.58601853961733426955667499597, −15.60346726377625189619725743561, −13.22411185504261166822431339568, −12.27468196176888365567476148924, −10.72985503783755615364940193829, −9.881724756107193033286377313124, −7.76587730927654606730829014073, −6.02389446783938231340212831717, −4.52431823521288325482668208631, −0.46657304302388910805689511729,
0.46657304302388910805689511729, 4.52431823521288325482668208631, 6.02389446783938231340212831717, 7.76587730927654606730829014073, 9.881724756107193033286377313124, 10.72985503783755615364940193829, 12.27468196176888365567476148924, 13.22411185504261166822431339568, 15.60346726377625189619725743561, 16.58601853961733426955667499597
