| L(s) = 1 | − 2.05·2-s + 2.20·4-s − 5-s + 4.21·7-s − 0.423·8-s + 2.05·10-s − 2.60·11-s + 0.414·13-s − 8.64·14-s − 3.54·16-s + 5.00·17-s − 4.86·19-s − 2.20·20-s + 5.34·22-s + 1.85·23-s + 25-s − 0.850·26-s + 9.30·28-s + 5.28·29-s + 0.229·31-s + 8.11·32-s − 10.2·34-s − 4.21·35-s − 3.21·37-s + 9.98·38-s + 0.423·40-s − 7.20·41-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.10·4-s − 0.447·5-s + 1.59·7-s − 0.149·8-s + 0.648·10-s − 0.786·11-s + 0.115·13-s − 2.31·14-s − 0.886·16-s + 1.21·17-s − 1.11·19-s − 0.493·20-s + 1.13·22-s + 0.386·23-s + 0.200·25-s − 0.166·26-s + 1.75·28-s + 0.980·29-s + 0.0412·31-s + 1.43·32-s − 1.76·34-s − 0.712·35-s − 0.529·37-s + 1.61·38-s + 0.0670·40-s − 1.12·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9226827660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9226827660\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.05T + 2T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 - 0.229T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 - 0.794T + 43T^{2} \) |
| 47 | \( 1 + 8.42T + 47T^{2} \) |
| 53 | \( 1 - 9.47T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 1.18T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.337385188680599734637096885261, −8.059815523712289034854831995998, −7.33469711630675225908727305790, −6.62744822900152741450213045247, −5.33489924350025931202161920482, −4.83454319981473764461433047382, −3.83961112992980669845129164442, −2.53434035366216295219600508393, −1.66233410362521498748820508937, −0.71900720786793808576667928665,
0.71900720786793808576667928665, 1.66233410362521498748820508937, 2.53434035366216295219600508393, 3.83961112992980669845129164442, 4.83454319981473764461433047382, 5.33489924350025931202161920482, 6.62744822900152741450213045247, 7.33469711630675225908727305790, 8.059815523712289034854831995998, 8.337385188680599734637096885261
