| L(s) = 1 | + 5-s + 5.12·7-s + 4·11-s − 1.12·13-s − 7.12·17-s + 7.12·19-s − 3.12·23-s + 25-s + 2·29-s + 5.12·31-s + 5.12·35-s − 5.12·37-s + 10.2·41-s − 4·43-s + 3.12·47-s + 19.2·49-s − 4.24·53-s + 4·55-s − 6.24·59-s + 6·61-s − 1.12·65-s − 8·67-s + 6.24·71-s + 8.24·73-s + 20.4·77-s + 6.87·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.93·7-s + 1.20·11-s − 0.311·13-s − 1.72·17-s + 1.63·19-s − 0.651·23-s + 0.200·25-s + 0.371·29-s + 0.920·31-s + 0.865·35-s − 0.842·37-s + 1.60·41-s − 0.609·43-s + 0.455·47-s + 2.74·49-s − 0.583·53-s + 0.539·55-s − 0.813·59-s + 0.768·61-s − 0.139·65-s − 0.977·67-s + 0.741·71-s + 0.965·73-s + 2.33·77-s + 0.773·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.147975296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.147975296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 6.87T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.122359927086181160272342408745, −7.45996360025344498794233633784, −6.72850411954182338407745959028, −5.96192894528235628719187508460, −5.07127723678723499096172500593, −4.59482573396637985572905968822, −3.86324922356266428134655286857, −2.56931065586988481134971726019, −1.79047949587263233835795232491, −1.03490118175612065436050553479,
1.03490118175612065436050553479, 1.79047949587263233835795232491, 2.56931065586988481134971726019, 3.86324922356266428134655286857, 4.59482573396637985572905968822, 5.07127723678723499096172500593, 5.96192894528235628719187508460, 6.72850411954182338407745959028, 7.45996360025344498794233633784, 8.122359927086181160272342408745
