| L(s) = 1 | + 2·3-s − 2·4-s − 3·5-s + 9-s − 4·12-s + 2·13-s − 6·15-s + 4·16-s + 2·19-s + 6·20-s − 23-s + 4·25-s − 4·27-s + 10·31-s − 2·36-s + 2·37-s + 4·39-s + 3·41-s + 7·43-s − 3·45-s − 6·47-s + 8·48-s − 4·52-s + 4·57-s − 6·59-s + 12·60-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.34·5-s + 1/3·9-s − 1.15·12-s + 0.554·13-s − 1.54·15-s + 16-s + 0.458·19-s + 1.34·20-s − 0.208·23-s + 4/5·25-s − 0.769·27-s + 1.79·31-s − 1/3·36-s + 0.328·37-s + 0.640·39-s + 0.468·41-s + 1.06·43-s − 0.447·45-s − 0.875·47-s + 1.15·48-s − 0.554·52-s + 0.529·57-s − 0.781·59-s + 1.54·60-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.72776980554250, −13.34906324789482, −12.78976164592662, −12.28899425738422, −11.84148889848608, −11.30656039187919, −10.85802065465565, −10.11601747828699, −9.674827115160550, −9.234875779258000, −8.659519717745710, −8.312185192142643, −7.969782392335181, −7.602079839558439, −6.985498194297140, −6.204376819858150, −5.688313787223408, −4.906863784214227, −4.430948023298448, −3.916659975874402, −3.592581556889271, −2.957463597711405, −2.518642704230849, −1.436900220904336, −0.7998684483141171, 0,
0.7998684483141171, 1.436900220904336, 2.518642704230849, 2.957463597711405, 3.592581556889271, 3.916659975874402, 4.430948023298448, 4.906863784214227, 5.688313787223408, 6.204376819858150, 6.985498194297140, 7.602079839558439, 7.969782392335181, 8.312185192142643, 8.659519717745710, 9.234875779258000, 9.674827115160550, 10.11601747828699, 10.85802065465565, 11.30656039187919, 11.84148889848608, 12.28899425738422, 12.78976164592662, 13.34906324789482, 13.72776980554250
