| L(s) = 1 | − 2.14·2-s + 2.61·4-s + 3.62·5-s + 2.48·7-s − 1.31·8-s − 7.77·10-s − 11-s + 0.181·13-s − 5.34·14-s − 2.40·16-s + 1.02·17-s + 8.45·19-s + 9.45·20-s + 2.14·22-s + 4.42·23-s + 8.11·25-s − 0.390·26-s + 6.49·28-s − 5.84·29-s + 9.14·31-s + 7.78·32-s − 2.20·34-s + 9.00·35-s + 1.08·37-s − 18.1·38-s − 4.74·40-s − 6.75·41-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.30·4-s + 1.61·5-s + 0.940·7-s − 0.463·8-s − 2.45·10-s − 0.301·11-s + 0.0504·13-s − 1.42·14-s − 0.601·16-s + 0.248·17-s + 1.93·19-s + 2.11·20-s + 0.457·22-s + 0.923·23-s + 1.62·25-s − 0.0765·26-s + 1.22·28-s − 1.08·29-s + 1.64·31-s + 1.37·32-s − 0.378·34-s + 1.52·35-s + 0.179·37-s − 2.94·38-s − 0.750·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680839894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680839894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 13 | \( 1 - 0.181T + 13T^{2} \) |
| 17 | \( 1 - 1.02T + 17T^{2} \) |
| 19 | \( 1 - 8.45T + 19T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 + 5.84T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 + 3.99T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 - 8.09T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.090123132944666054798377693365, −7.64295867130059021039061061580, −6.83797479199931628634405462554, −6.12223403679161584880310711595, −5.19372586108928515180321619220, −4.86361097701548535529015927654, −3.23322997426721006846135601387, −2.33910750610119965264691110360, −1.54993844702240346956756008628, −0.961207751784795193313491134735,
0.961207751784795193313491134735, 1.54993844702240346956756008628, 2.33910750610119965264691110360, 3.23322997426721006846135601387, 4.86361097701548535529015927654, 5.19372586108928515180321619220, 6.12223403679161584880310711595, 6.83797479199931628634405462554, 7.64295867130059021039061061580, 8.090123132944666054798377693365
