| L(s) = 1 | + 0.0583·2-s − 1.99·4-s − 3.92·5-s − 2.69·7-s − 0.233·8-s − 0.228·10-s + 3.25·11-s + 0.958·13-s − 0.157·14-s + 3.97·16-s + 1.23·17-s − 19-s + 7.83·20-s + 0.189·22-s − 3.54·23-s + 10.3·25-s + 0.0559·26-s + 5.37·28-s − 4.36·29-s − 9.98·31-s + 0.698·32-s + 0.0723·34-s + 10.5·35-s − 7.14·37-s − 0.0583·38-s + 0.914·40-s − 5.14·41-s + ⋯ |
| L(s) = 1 | + 0.0412·2-s − 0.998·4-s − 1.75·5-s − 1.01·7-s − 0.0824·8-s − 0.0723·10-s + 0.981·11-s + 0.265·13-s − 0.0419·14-s + 0.994·16-s + 0.300·17-s − 0.229·19-s + 1.75·20-s + 0.0404·22-s − 0.739·23-s + 2.07·25-s + 0.0109·26-s + 1.01·28-s − 0.810·29-s − 1.79·31-s + 0.123·32-s + 0.0123·34-s + 1.78·35-s − 1.17·37-s − 0.00946·38-s + 0.144·40-s − 0.804·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2310092258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2310092258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0583T + 2T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 0.958T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 + 3.54T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 + 7.14T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 2.11T + 43T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 + 9.08T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 - 7.28T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 4.17T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 + 1.85T + 97T^{2} \) |
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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
−7.84838440904989701876195444143, −7.25498084917290670508733659439, −6.53177267364417307654984291981, −5.72769048580078404063277373585, −4.84900460007460600033804569684, −4.03120406350282575484877023961, −3.65234895685489190938885030175, −3.20708896434403921993823843912, −1.52755328021107544405518214208, −0.24488569334956094280553337215,
0.24488569334956094280553337215, 1.52755328021107544405518214208, 3.20708896434403921993823843912, 3.65234895685489190938885030175, 4.03120406350282575484877023961, 4.84900460007460600033804569684, 5.72769048580078404063277373585, 6.53177267364417307654984291981, 7.25498084917290670508733659439, 7.84838440904989701876195444143
