| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 2.12·11-s − 6.68·13-s + 14-s + 16-s + 5·17-s − 3.56·19-s + 20-s + 2.12·22-s + 2.43·23-s + 25-s + 6.68·26-s − 28-s − 5.68·29-s + 0.561·31-s − 32-s − 5·34-s − 35-s + 37-s + 3.56·38-s − 40-s + 3.68·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.640·11-s − 1.85·13-s + 0.267·14-s + 0.250·16-s + 1.21·17-s − 0.817·19-s + 0.223·20-s + 0.452·22-s + 0.508·23-s + 0.200·25-s + 1.31·26-s − 0.188·28-s − 1.05·29-s + 0.100·31-s − 0.176·32-s − 0.857·34-s − 0.169·35-s + 0.164·37-s + 0.577·38-s − 0.158·40-s + 0.575·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030161308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030161308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 6.68T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 0.561T + 31T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 + 0.315T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 6.68T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.715883256715584154895621343962, −7.77307716560079206746092467351, −7.33156246421762116416402557818, −6.54773560955918072875671895089, −5.56539404272364765377855094835, −5.06064535762081458263344268527, −3.81376370844217858271393408054, −2.69123950864686021440639158442, −2.13234434297519863382194167252, −0.64505600146968484497589973860,
0.64505600146968484497589973860, 2.13234434297519863382194167252, 2.69123950864686021440639158442, 3.81376370844217858271393408054, 5.06064535762081458263344268527, 5.56539404272364765377855094835, 6.54773560955918072875671895089, 7.33156246421762116416402557818, 7.77307716560079206746092467351, 8.715883256715584154895621343962
