| L(s) = 1 | − 2-s − 3-s − 4-s − 3·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 3·10-s − 2·11-s + 12-s + 3·13-s + 14-s + 3·15-s − 16-s − 6·17-s + 2·18-s + 4·19-s + 3·20-s + 21-s + 2·22-s + 2·23-s − 3·24-s + 4·25-s − 3·26-s + 5·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.948·10-s − 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.774·15-s − 1/4·16-s − 1.45·17-s + 0.471·18-s + 0.917·19-s + 0.670·20-s + 0.218·21-s + 0.426·22-s + 0.417·23-s − 0.612·24-s + 4/5·25-s − 0.588·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 79 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.76412543794758860714897963906, −12.72651837214302869531384269103, −11.35676198529650461404936978282, −10.80850123926612739376478567978, −9.181129339396648324794848537589, −8.282431792555002607496667489786, −7.12147698344619796195085066396, −5.29917071437467895295333587611, −3.76592143771251579951082054976, 0,
3.76592143771251579951082054976, 5.29917071437467895295333587611, 7.12147698344619796195085066396, 8.282431792555002607496667489786, 9.181129339396648324794848537589, 10.80850123926612739376478567978, 11.35676198529650461404936978282, 12.72651837214302869531384269103, 13.76412543794758860714897963906
