| L(s) = 1 | + 3-s + 0.801·5-s − 3.69·7-s + 9-s + 2.85·11-s + 0.801·15-s + 4.44·17-s − 2.26·19-s − 3.69·21-s − 7.78·23-s − 4.35·25-s + 27-s + 0.246·29-s + 0.466·31-s + 2.85·33-s − 2.96·35-s − 6.65·37-s + 8.63·41-s + 9.76·43-s + 0.801·45-s − 11.8·47-s + 6.63·49-s + 4.44·51-s − 8.94·53-s + 2.28·55-s − 2.26·57-s − 0.396·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.358·5-s − 1.39·7-s + 0.333·9-s + 0.859·11-s + 0.207·15-s + 1.07·17-s − 0.520·19-s − 0.805·21-s − 1.62·23-s − 0.871·25-s + 0.192·27-s + 0.0458·29-s + 0.0838·31-s + 0.496·33-s − 0.500·35-s − 1.09·37-s + 1.34·41-s + 1.48·43-s + 0.119·45-s − 1.73·47-s + 0.947·49-s + 0.622·51-s − 1.22·53-s + 0.308·55-s − 0.300·57-s − 0.0515·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.801T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 - 0.246T + 29T^{2} \) |
| 31 | \( 1 - 0.466T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 - 9.76T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 0.396T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 7.77T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.53764303698918253423062789278, −6.70282113612820539352966502732, −6.11388624621262911548689039401, −5.67947527671296806352073129438, −4.40805526970759398381150640910, −3.76997444682107814486918674554, −3.18252479493396628000384013744, −2.28808856056059731250956843147, −1.38623690374588204164760008827, 0,
1.38623690374588204164760008827, 2.28808856056059731250956843147, 3.18252479493396628000384013744, 3.76997444682107814486918674554, 4.40805526970759398381150640910, 5.67947527671296806352073129438, 6.11388624621262911548689039401, 6.70282113612820539352966502732, 7.53764303698918253423062789278
