| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 4·13-s − 15-s + 4·17-s + 4·19-s − 4·21-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 4·35-s − 2·37-s + 4·39-s + 10·41-s − 43-s + 45-s − 4·47-s + 9·49-s − 4·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 0.258·15-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.152·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s − 0.560·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.95523998001325, −14.35493208944675, −14.11956261585125, −13.53773301590016, −12.82759279783551, −12.31994791126439, −11.88460423773833, −11.39829648478579, −10.80207067330897, −10.36328058559427, −9.702875956444938, −9.531607069743465, −8.381658224588728, −8.029467005193662, −7.467761612868208, −7.231389979487450, −6.022281500521002, −5.735320222992554, −5.108543799526496, −4.793950280464288, −4.089530364060287, −3.146079291836175, −2.339419434435542, −1.834076165264527, −1.021694960584586, 0,
1.021694960584586, 1.834076165264527, 2.339419434435542, 3.146079291836175, 4.089530364060287, 4.793950280464288, 5.108543799526496, 5.735320222992554, 6.022281500521002, 7.231389979487450, 7.467761612868208, 8.029467005193662, 8.381658224588728, 9.531607069743465, 9.702875956444938, 10.36328058559427, 10.80207067330897, 11.39829648478579, 11.88460423773833, 12.31994791126439, 12.82759279783551, 13.53773301590016, 14.11956261585125, 14.35493208944675, 14.95523998001325
