| L(s) = 1 | − 2.51·2-s + 4.32·4-s + 5-s − 2.83·7-s − 5.84·8-s − 2.51·10-s + 1.32·11-s − 5.12·13-s + 7.11·14-s + 6.04·16-s − 4.61·17-s + 1.84·19-s + 4.32·20-s − 3.34·22-s + 2.29·23-s + 25-s + 12.8·26-s − 12.2·28-s − 0.167·29-s + 9.85·31-s − 3.51·32-s + 11.5·34-s − 2.83·35-s − 0.207·37-s − 4.64·38-s − 5.84·40-s − 5.20·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.16·4-s + 0.447·5-s − 1.06·7-s − 2.06·8-s − 0.795·10-s + 0.400·11-s − 1.42·13-s + 1.90·14-s + 1.51·16-s − 1.11·17-s + 0.423·19-s + 0.966·20-s − 0.712·22-s + 0.477·23-s + 0.200·25-s + 2.52·26-s − 2.31·28-s − 0.0310·29-s + 1.76·31-s − 0.622·32-s + 1.98·34-s − 0.478·35-s − 0.0341·37-s − 0.753·38-s − 0.923·40-s − 0.812·41-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 | \( 1 + 2.51T + 2T^{2} \) |
| 7 | \( 1 + 2.83T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 + 0.167T + 29T^{2} \) |
| 31 | \( 1 - 9.85T + 31T^{2} \) |
| 37 | \( 1 + 0.207T + 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 - 3.11T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.69T + 67T^{2} \) |
| 71 | \( 1 + 0.991T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 97 | \( 1 - 1.80T + 97T^{2} \) |
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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
−8.306272688669127696664357610112, −7.31829655384236861795594683182, −6.86133356013752849553906966846, −6.33915629412473082872898439942, −5.32497909513918770934889478380, −4.17800599529065440761723392238, −2.77588875433909425430119883925, −2.40029136759666425083287819783, −1.09918351396455551656916745411, 0,
1.09918351396455551656916745411, 2.40029136759666425083287819783, 2.77588875433909425430119883925, 4.17800599529065440761723392238, 5.32497909513918770934889478380, 6.33915629412473082872898439942, 6.86133356013752849553906966846, 7.31829655384236861795594683182, 8.306272688669127696664357610112
