| L(s) = 1 | − 2.03·2-s + 2.12·4-s + 3.82·5-s + 7-s − 0.255·8-s − 7.77·10-s + 5.96·11-s + 1.44·13-s − 2.03·14-s − 3.73·16-s − 6.06·17-s − 0.0743·19-s + 8.13·20-s − 12.1·22-s + 4.43·23-s + 9.64·25-s − 2.93·26-s + 2.12·28-s + 1.92·29-s + 1.76·31-s + 8.09·32-s + 12.3·34-s + 3.82·35-s + 0.497·37-s + 0.150·38-s − 0.978·40-s + 41-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.06·4-s + 1.71·5-s + 0.377·7-s − 0.0903·8-s − 2.45·10-s + 1.79·11-s + 0.400·13-s − 0.542·14-s − 0.933·16-s − 1.47·17-s − 0.0170·19-s + 1.81·20-s − 2.58·22-s + 0.924·23-s + 1.92·25-s − 0.575·26-s + 0.401·28-s + 0.357·29-s + 0.316·31-s + 1.43·32-s + 2.11·34-s + 0.646·35-s + 0.0817·37-s + 0.0244·38-s − 0.154·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498848299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498848299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 6.06T + 17T^{2} \) |
| 19 | \( 1 + 0.0743T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 0.497T + 37T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 + 5.87T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.670T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.021175805798888560331719358357, −8.588212051734881554262611485874, −7.40483279238981171562670950505, −6.46533915833146399622509770148, −6.35019881710071643454591387740, −5.04516550219879289519868737073, −4.15862689384707400700842095255, −2.60069117858613596252637627864, −1.71206049816177495705416488967, −1.08044488103993797164343954508,
1.08044488103993797164343954508, 1.71206049816177495705416488967, 2.60069117858613596252637627864, 4.15862689384707400700842095255, 5.04516550219879289519868737073, 6.35019881710071643454591387740, 6.46533915833146399622509770148, 7.40483279238981171562670950505, 8.588212051734881554262611485874, 9.021175805798888560331719358357
