| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 11-s − 4·13-s + 14-s + 16-s + 2·17-s − 6·19-s + 2·20-s + 22-s − 2·23-s − 25-s + 4·26-s − 28-s − 2·29-s − 8·31-s − 32-s − 2·34-s − 2·35-s − 2·37-s + 6·38-s − 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.301·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.447·20-s + 0.213·22-s − 0.417·23-s − 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.338·35-s − 0.328·37-s + 0.973·38-s − 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.286627858952939688643238165404, −8.498661052564373095840327857913, −7.55983219604555693601523262065, −6.82418002670887555935696009359, −5.92515700044966766081550832122, −5.21358330310255541751497667932, −3.88305920202674694971898012431, −2.57760145046844813266140965153, −1.81668054985529005187136744964, 0,
1.81668054985529005187136744964, 2.57760145046844813266140965153, 3.88305920202674694971898012431, 5.21358330310255541751497667932, 5.92515700044966766081550832122, 6.82418002670887555935696009359, 7.55983219604555693601523262065, 8.498661052564373095840327857913, 9.286627858952939688643238165404
