| L(s) = 1 | + 2-s − 3.23·3-s + 4-s − 1.14·5-s − 3.23·6-s + 8-s + 7.46·9-s − 1.14·10-s − 0.850·11-s − 3.23·12-s − 3.66·13-s + 3.70·15-s + 16-s − 0.194·17-s + 7.46·18-s + 6.99·19-s − 1.14·20-s − 0.850·22-s − 3.46·23-s − 3.23·24-s − 3.69·25-s − 3.66·26-s − 14.4·27-s − 29-s + 3.70·30-s + 3.97·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s − 0.511·5-s − 1.32·6-s + 0.353·8-s + 2.48·9-s − 0.361·10-s − 0.256·11-s − 0.933·12-s − 1.01·13-s + 0.955·15-s + 0.250·16-s − 0.0471·17-s + 1.75·18-s + 1.60·19-s − 0.255·20-s − 0.181·22-s − 0.721·23-s − 0.660·24-s − 0.738·25-s − 0.718·26-s − 2.77·27-s − 0.185·29-s + 0.675·30-s + 0.713·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 11 | \( 1 + 0.850T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 + 0.194T + 17T^{2} \) |
| 19 | \( 1 - 6.99T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 - 7.20T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 + 0.0538T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 8.04T + 59T^{2} \) |
| 61 | \( 1 - 5.08T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.31T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−7.979512776218004828345084265059, −7.37029426501220590225071278389, −6.79154046246677045827208656811, −5.75760327735358633243494712444, −5.49309309228463545295925437027, −4.55402668345791104712431028735, −4.04321432478185671176907791488, −2.70915382162009503824310560576, −1.27612471296597777331642573794, 0,
1.27612471296597777331642573794, 2.70915382162009503824310560576, 4.04321432478185671176907791488, 4.55402668345791104712431028735, 5.49309309228463545295925437027, 5.75760327735358633243494712444, 6.79154046246677045827208656811, 7.37029426501220590225071278389, 7.979512776218004828345084265059
