| L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s − 2·12-s − 4·13-s − 15-s + 4·16-s − 3·17-s − 19-s + 2·20-s − 21-s − 23-s + 25-s + 27-s + 2·28-s + 29-s + 8·31-s + 35-s − 2·36-s + 8·37-s − 4·39-s + 6·41-s + 8·43-s − 45-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.229·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.185·29-s + 1.43·31-s + 0.169·35-s − 1/3·36-s + 1.31·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−17.05944623764172, −16.23639581088657, −15.76297860205183, −15.12318926384466, −14.48514076277824, −14.20508206811226, −13.45816752736327, −12.85877274353898, −12.55820065822204, −11.80114082267408, −11.09149064935827, −10.22973081343916, −9.797024390434613, −9.202665094125352, −8.699470362421521, −7.914573725381631, −7.603486506643287, −6.674614533184266, −5.991580556058792, −5.024067701189357, −4.388803927462127, −4.009012678168782, −2.952983101680841, −2.420697105985141, −1.032706850182010, 0,
1.032706850182010, 2.420697105985141, 2.952983101680841, 4.009012678168782, 4.388803927462127, 5.024067701189357, 5.991580556058792, 6.674614533184266, 7.603486506643287, 7.914573725381631, 8.699470362421521, 9.202665094125352, 9.797024390434613, 10.22973081343916, 11.09149064935827, 11.80114082267408, 12.55820065822204, 12.85877274353898, 13.45816752736327, 14.20508206811226, 14.48514076277824, 15.12318926384466, 15.76297860205183, 16.23639581088657, 17.05944623764172
