| L(s) = 1 | − 1.37·2-s − 0.110·4-s + 5-s − 2.28·7-s + 2.90·8-s − 1.37·10-s − 3.97·11-s + 6.54·13-s + 3.14·14-s − 3.76·16-s − 3.55·17-s + 7.14·19-s − 0.110·20-s + 5.46·22-s + 3.65·23-s + 25-s − 9.00·26-s + 0.252·28-s + 4.70·29-s + 1.10·31-s − 0.623·32-s + 4.88·34-s − 2.28·35-s − 7.55·37-s − 9.81·38-s + 2.90·40-s − 7.27·41-s + ⋯ |
| L(s) = 1 | − 0.972·2-s − 0.0551·4-s + 0.447·5-s − 0.865·7-s + 1.02·8-s − 0.434·10-s − 1.19·11-s + 1.81·13-s + 0.841·14-s − 0.941·16-s − 0.861·17-s + 1.63·19-s − 0.0246·20-s + 1.16·22-s + 0.763·23-s + 0.200·25-s − 1.76·26-s + 0.0477·28-s + 0.873·29-s + 0.198·31-s − 0.110·32-s + 0.836·34-s − 0.386·35-s − 1.24·37-s − 1.59·38-s + 0.458·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9151489053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9151489053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 7.14T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 + 7.27T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 7.36T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−8.504220513009613345681490832238, −8.005289293296928128275176203881, −6.99513529842658279783392207591, −6.47822912145885719097331883123, −5.45714531973491345848319078689, −4.85836061642217408151399135547, −3.63812086935794203305079526574, −2.92694882620849883926915353702, −1.66257688411223633260727578422, −0.66058549390478233080685299563,
0.66058549390478233080685299563, 1.66257688411223633260727578422, 2.92694882620849883926915353702, 3.63812086935794203305079526574, 4.85836061642217408151399135547, 5.45714531973491345848319078689, 6.47822912145885719097331883123, 6.99513529842658279783392207591, 8.005289293296928128275176203881, 8.504220513009613345681490832238
