| L(s) = 1 | + 2-s + 4-s − 5-s + 1.20·7-s + 8-s − 10-s + 11-s − 2.15·13-s + 1.20·14-s + 16-s − 1.04·17-s − 2.04·19-s − 20-s + 22-s − 1.95·23-s + 25-s − 2.15·26-s + 1.20·28-s − 7.65·29-s + 4.20·31-s + 32-s − 1.04·34-s − 1.20·35-s + 8.49·37-s − 2.04·38-s − 40-s − 6.24·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.454·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.597·13-s + 0.321·14-s + 0.250·16-s − 0.254·17-s − 0.469·19-s − 0.223·20-s + 0.213·22-s − 0.407·23-s + 0.200·25-s − 0.422·26-s + 0.227·28-s − 1.42·29-s + 0.754·31-s + 0.176·32-s − 0.179·34-s − 0.203·35-s + 1.39·37-s − 0.332·38-s − 0.158·40-s − 0.975·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 1.20T + 7T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 + 1.95T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 4.04T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 9.45T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.54T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.51844676688645448208064550531, −6.53339424932760005647593227381, −6.12839611693970925324147035948, −5.10482530455309238692425913172, −4.64961602116588401167709314080, −3.93909719690796558534135756456, −3.19000925855323457943119600477, −2.27853235096089070592984063135, −1.45785899976218745517282714953, 0,
1.45785899976218745517282714953, 2.27853235096089070592984063135, 3.19000925855323457943119600477, 3.93909719690796558534135756456, 4.64961602116588401167709314080, 5.10482530455309238692425913172, 6.12839611693970925324147035948, 6.53339424932760005647593227381, 7.51844676688645448208064550531
