| L(s) = 1 | + 1.81·2-s + 1.28·4-s + 5-s − 0.549·7-s − 1.30·8-s + 1.81·10-s − 4.46·11-s + 5.90·13-s − 0.994·14-s − 4.92·16-s − 7.44·17-s − 6.48·19-s + 1.28·20-s − 8.09·22-s + 8.93·23-s + 25-s + 10.7·26-s − 0.703·28-s + 2.29·29-s − 5.81·31-s − 6.30·32-s − 13.4·34-s − 0.549·35-s + 10.4·37-s − 11.7·38-s − 1.30·40-s − 7.73·41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.640·4-s + 0.447·5-s − 0.207·7-s − 0.460·8-s + 0.572·10-s − 1.34·11-s + 1.63·13-s − 0.265·14-s − 1.23·16-s − 1.80·17-s − 1.48·19-s + 0.286·20-s − 1.72·22-s + 1.86·23-s + 0.200·25-s + 2.09·26-s − 0.132·28-s + 0.426·29-s − 1.04·31-s − 1.11·32-s − 2.31·34-s − 0.0928·35-s + 1.71·37-s − 1.90·38-s − 0.205·40-s − 1.20·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.81T + 2T^{2} \) |
| 7 | \( 1 + 0.549T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 17 | \( 1 + 7.44T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 2.23T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 + 0.236T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 9.22T + 83T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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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
−8.240381897896727542581665570173, −6.89750838208549741093604240692, −6.41616511615426566768858740393, −5.83968235020942359242440355545, −4.86932794589156470763684972350, −4.51411970691220010763408559819, −3.41593927254978689080182945514, −2.77745840864892817917699739604, −1.80551479099797230059874769607, 0,
1.80551479099797230059874769607, 2.77745840864892817917699739604, 3.41593927254978689080182945514, 4.51411970691220010763408559819, 4.86932794589156470763684972350, 5.83968235020942359242440355545, 6.41616511615426566768858740393, 6.89750838208549741093604240692, 8.240381897896727542581665570173
