| L(s) = 1 | − 3.14·2-s − 118.·4-s + 39.2·5-s + 434.·7-s + 774.·8-s − 123.·10-s + 6.39e3·11-s + 442.·13-s − 1.36e3·14-s + 1.26e4·16-s + 2.63e4·17-s + 7.74e3·19-s − 4.63e3·20-s − 2.01e4·22-s + 5.86e4·23-s − 7.65e4·25-s − 1.39e3·26-s − 5.13e4·28-s − 2.70e4·29-s + 1.00e5·31-s − 1.39e5·32-s − 8.28e4·34-s + 1.70e4·35-s + 3.58e5·37-s − 2.44e4·38-s + 3.04e4·40-s + 7.43e4·41-s + ⋯ |
| L(s) = 1 | − 0.278·2-s − 0.922·4-s + 0.140·5-s + 0.478·7-s + 0.535·8-s − 0.0391·10-s + 1.44·11-s + 0.0559·13-s − 0.133·14-s + 0.773·16-s + 1.29·17-s + 0.259·19-s − 0.129·20-s − 0.403·22-s + 1.00·23-s − 0.980·25-s − 0.0155·26-s − 0.441·28-s − 0.205·29-s + 0.605·31-s − 0.750·32-s − 0.361·34-s + 0.0672·35-s + 1.16·37-s − 0.0721·38-s + 0.0752·40-s + 0.168·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.084584650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084584650\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 3.14T + 128T^{2} \) |
| 5 | \( 1 - 39.2T + 7.81e4T^{2} \) |
| 7 | \( 1 - 434.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 442.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 7.74e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.70e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.43e4T + 1.94e11T^{2} \) |
| 47 | \( 1 + 4.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.17e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.70e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.64e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.99e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.93e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.24e7T + 8.07e13T^{2} \) |
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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
−9.774036725426496917598943855101, −9.403803031550446407952062595454, −8.342069110949654978662201862499, −7.59335732221050738387226482957, −6.31528407284447766638231197293, −5.25082827352185142505291733871, −4.28138306272018877223526874996, −3.31737288029826516347346479208, −1.54940825471449487430146174455, −0.78655006723050280498857724439,
0.78655006723050280498857724439, 1.54940825471449487430146174455, 3.31737288029826516347346479208, 4.28138306272018877223526874996, 5.25082827352185142505291733871, 6.31528407284447766638231197293, 7.59335732221050738387226482957, 8.342069110949654978662201862499, 9.403803031550446407952062595454, 9.774036725426496917598943855101
