L(s) = 1 | + 2-s + 1.81·3-s + 4-s + 1.81·6-s + 2.52·7-s + 8-s + 0.289·9-s − 2.91·11-s + 1.81·12-s + 3.10·13-s + 2.52·14-s + 16-s + 6.91·17-s + 0.289·18-s + 1.81·19-s + 4.57·21-s − 2.91·22-s − 5.68·23-s + 1.81·24-s + 3.10·26-s − 4.91·27-s + 2.52·28-s + 29-s − 8.12·31-s + 32-s − 5.28·33-s + 6.91·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.740·6-s + 0.954·7-s + 0.353·8-s + 0.0963·9-s − 0.879·11-s + 0.523·12-s + 0.860·13-s + 0.674·14-s + 0.250·16-s + 1.67·17-s + 0.0681·18-s + 0.416·19-s + 0.999·21-s − 0.621·22-s − 1.18·23-s + 0.370·24-s + 0.608·26-s − 0.946·27-s + 0.477·28-s + 0.185·29-s − 1.45·31-s + 0.176·32-s − 0.920·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.057627778\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.057627778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 5.04T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 - 5.30T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 5.23T + 89T^{2} \) |
| 97 | \( 1 - 1.27T + 97T^{2} \) |
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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
−9.476933673187974219841409906785, −8.487817834420716056606011191789, −7.83942727880498992516297945912, −7.46807759906449086063830289705, −5.89349271355209366402662140182, −5.46468383966444314028411366713, −4.27189154369678671808371603153, −3.43725222673329893577951717217, −2.58088425728569898981368948280, −1.49224604013497417637561469512,
1.49224604013497417637561469512, 2.58088425728569898981368948280, 3.43725222673329893577951717217, 4.27189154369678671808371603153, 5.46468383966444314028411366713, 5.89349271355209366402662140182, 7.46807759906449086063830289705, 7.83942727880498992516297945912, 8.487817834420716056606011191789, 9.476933673187974219841409906785