| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 4·7-s − 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 6·13-s + 4·14-s + 15-s − 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 4·21-s − 4·22-s − 4·23-s − 3·24-s + 25-s + 6·26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.385999425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385999425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.08632533794567075438500598760, −10.36706411410877680974948279833, −9.180815844797497435996582347463, −8.285142698379180989401958892533, −7.82152461814686983756055018114, −6.04599499864174651869297996238, −5.30506594738975870887892595526, −4.31801287744868728937060903572, −3.22174431616232965676904645612, −1.68581285052697885373760198065,
1.68581285052697885373760198065, 3.22174431616232965676904645612, 4.31801287744868728937060903572, 5.30506594738975870887892595526, 6.04599499864174651869297996238, 7.82152461814686983756055018114, 8.285142698379180989401958892533, 9.180815844797497435996582347463, 10.36706411410877680974948279833, 11.08632533794567075438500598760
