| L(s) = 1 | + 2-s − 4-s − 3·8-s − 11-s + 13-s − 16-s + 4·17-s − 8·19-s − 22-s − 5·25-s + 26-s − 4·29-s − 6·31-s + 5·32-s + 4·34-s − 6·37-s − 8·38-s − 6·41-s − 2·43-s + 44-s + 8·47-s − 7·49-s − 5·50-s − 52-s − 6·53-s − 4·58-s − 14·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.301·11-s + 0.277·13-s − 1/4·16-s + 0.970·17-s − 1.83·19-s − 0.213·22-s − 25-s + 0.196·26-s − 0.742·29-s − 1.07·31-s + 0.883·32-s + 0.685·34-s − 0.986·37-s − 1.29·38-s − 0.937·41-s − 0.304·43-s + 0.150·44-s + 1.16·47-s − 49-s − 0.707·50-s − 0.138·52-s − 0.824·53-s − 0.525·58-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.220863465579452413866170833774, −8.474154558304973608799593431124, −7.70197854786009233833355490486, −6.52814972783847237644052290259, −5.75387120885855194840916483608, −5.00796137364139601993303055866, −4.01003994013312318326579777795, −3.33349505016483039001926303041, −1.94040075234719115377357472617, 0,
1.94040075234719115377357472617, 3.33349505016483039001926303041, 4.01003994013312318326579777795, 5.00796137364139601993303055866, 5.75387120885855194840916483608, 6.52814972783847237644052290259, 7.70197854786009233833355490486, 8.474154558304973608799593431124, 9.220863465579452413866170833774
