| L(s) = 1 | − 2·2-s + 4·4-s − 2.36·5-s − 18.5·7-s − 8·8-s + 4.72·10-s + 15.3·11-s + 27.8·13-s + 37.0·14-s + 16·16-s + 62.4·17-s + 55.3·19-s − 9.44·20-s − 30.7·22-s − 44.3·23-s − 119.·25-s − 55.6·26-s − 74.0·28-s − 29·29-s − 207.·31-s − 32·32-s − 124.·34-s + 43.6·35-s + 303.·37-s − 110.·38-s + 18.8·40-s − 125.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.211·5-s − 0.999·7-s − 0.353·8-s + 0.149·10-s + 0.421·11-s + 0.593·13-s + 0.706·14-s + 0.250·16-s + 0.890·17-s + 0.668·19-s − 0.105·20-s − 0.298·22-s − 0.401·23-s − 0.955·25-s − 0.419·26-s − 0.499·28-s − 0.185·29-s − 1.19·31-s − 0.176·32-s − 0.629·34-s + 0.210·35-s + 1.34·37-s − 0.472·38-s + 0.0746·40-s − 0.478·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 5 | \( 1 + 2.36T + 125T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 11 | \( 1 - 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.3T + 1.21e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 50.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 692.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 809.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 749.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 54.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 184.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 752.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 902.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 953.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−9.713988494049806162336174324219, −9.407390655746228931128102630610, −8.202265356605173528223512409410, −7.43089661223318337287165644663, −6.40482560135211149216664033269, −5.62347541953503055320372047919, −3.93405933755530498601890259173, −3.02865890544754311347787824013, −1.42923750365593147893384171056, 0,
1.42923750365593147893384171056, 3.02865890544754311347787824013, 3.93405933755530498601890259173, 5.62347541953503055320372047919, 6.40482560135211149216664033269, 7.43089661223318337287165644663, 8.202265356605173528223512409410, 9.407390655746228931128102630610, 9.713988494049806162336174324219
