| L(s) = 1 | + 0.319·5-s − 2.61·7-s + 4.31·11-s + 6.51·13-s − 4.19·17-s − 19-s − 0.639·23-s − 4.89·25-s + 7.87·29-s − 6·31-s − 0.837·35-s + 4·37-s + 12.5·41-s − 1.38·43-s + 7.55·47-s − 0.137·49-s + 13.1·53-s + 1.38·55-s + 13.0·59-s − 5.89·61-s + 2.08·65-s + 11.7·67-s − 11.7·71-s + 15.1·73-s − 11.3·77-s + 0.517·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | + 0.142·5-s − 0.990·7-s + 1.30·11-s + 1.80·13-s − 1.01·17-s − 0.229·19-s − 0.133·23-s − 0.979·25-s + 1.46·29-s − 1.07·31-s − 0.141·35-s + 0.657·37-s + 1.95·41-s − 0.210·43-s + 1.10·47-s − 0.0196·49-s + 1.80·53-s + 0.186·55-s + 1.69·59-s − 0.755·61-s + 0.258·65-s + 1.43·67-s − 1.39·71-s + 1.77·73-s − 1.28·77-s + 0.0582·79-s + 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710224413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710224413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.319T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 23 | \( 1 + 0.639T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 0.517T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−9.359602314342669741323670576344, −8.991834534655780255362601413591, −8.120940837280305430405524871748, −6.85312968774479620215852630642, −6.36343209766069266682641200469, −5.70600069853251981828165654330, −4.11612555867565220099038335731, −3.74719719000105245813576785221, −2.38838237708514398853390894361, −0.992702100294505427113430246966,
0.992702100294505427113430246966, 2.38838237708514398853390894361, 3.74719719000105245813576785221, 4.11612555867565220099038335731, 5.70600069853251981828165654330, 6.36343209766069266682641200469, 6.85312968774479620215852630642, 8.120940837280305430405524871748, 8.991834534655780255362601413591, 9.359602314342669741323670576344
