| L(s) = 1 | − 0.258·2-s − 7.93·4-s + 5·5-s + 14.5·7-s + 4.12·8-s − 1.29·10-s − 49.2·11-s + 72.1·13-s − 3.75·14-s + 62.3·16-s + 118.·17-s + 123.·19-s − 39.6·20-s + 12.7·22-s − 91.4·23-s + 25·25-s − 18.6·26-s − 115.·28-s + 174.·29-s − 46.2·31-s − 49.1·32-s − 30.5·34-s + 72.5·35-s + 154.·37-s − 31.9·38-s + 20.6·40-s − 364.·41-s + ⋯ |
| L(s) = 1 | − 0.0914·2-s − 0.991·4-s + 0.447·5-s + 0.783·7-s + 0.182·8-s − 0.0409·10-s − 1.35·11-s + 1.53·13-s − 0.0716·14-s + 0.974·16-s + 1.68·17-s + 1.48·19-s − 0.443·20-s + 0.123·22-s − 0.829·23-s + 0.200·25-s − 0.140·26-s − 0.777·28-s + 1.11·29-s − 0.268·31-s − 0.271·32-s − 0.154·34-s + 0.350·35-s + 0.688·37-s − 0.136·38-s + 0.0814·40-s − 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.507343909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507343909\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 0.258T + 8T^{2} \) |
| 7 | \( 1 - 14.5T + 343T^{2} \) |
| 11 | \( 1 + 49.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 13.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 76.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 335.T + 9.12e5T^{2} \) |
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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
−12.97725667584750440774115446170, −11.76637858324474675292128512328, −10.47857349900496496955351377503, −9.740625732786603178422860955747, −8.387113389414034101231407143757, −7.79568480494185132574097298263, −5.78115447342691288264749105877, −4.99528109092923355629027008038, −3.37523131570203269400819655465, −1.16991633633344706421380871163,
1.16991633633344706421380871163, 3.37523131570203269400819655465, 4.99528109092923355629027008038, 5.78115447342691288264749105877, 7.79568480494185132574097298263, 8.387113389414034101231407143757, 9.740625732786603178422860955747, 10.47857349900496496955351377503, 11.76637858324474675292128512328, 12.97725667584750440774115446170
