| L(s) = 1 | − 2-s + 4-s + 4·5-s − 4·7-s − 8-s − 3·9-s − 4·10-s + 2·11-s − 2·13-s + 4·14-s + 16-s − 2·17-s + 3·18-s − 2·19-s + 4·20-s − 2·22-s + 23-s + 11·25-s + 2·26-s − 4·28-s + 2·29-s − 32-s + 2·34-s − 16·35-s − 3·36-s − 4·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 1.51·7-s − 0.353·8-s − 9-s − 1.26·10-s + 0.603·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.894·20-s − 0.426·22-s + 0.208·23-s + 11/5·25-s + 0.392·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s − 2.70·35-s − 1/2·36-s − 0.657·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6609041113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6609041113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.21405768024039301547526001930, −14.59615890156611387959587952197, −13.53832745679506300332677090632, −12.42275899419421956395967188360, −10.68153496695016269554639136746, −9.594811427439617650906242640583, −8.971935452602447588346290189052, −6.69441913641732873591204403077, −5.84432417959727908710252168629, −2.63232737474552247848840602025,
2.63232737474552247848840602025, 5.84432417959727908710252168629, 6.69441913641732873591204403077, 8.971935452602447588346290189052, 9.594811427439617650906242640583, 10.68153496695016269554639136746, 12.42275899419421956395967188360, 13.53832745679506300332677090632, 14.59615890156611387959587952197, 16.21405768024039301547526001930
