| L(s) = 1 | − 0.418·7-s + 6.50·11-s + 2.62·13-s + 2.27·17-s + 4.27·19-s − 4.27·23-s − 2.62·29-s + 3·31-s + 6.09·37-s + 8.71·41-s − 10.3·43-s + 10.5·47-s − 6.82·49-s − 5.54·53-s − 3.46·59-s + 10.2·61-s − 6.09·67-s − 2.62·71-s + 6.50·73-s − 2.72·77-s + 6.27·79-s + 0.450·83-s − 10.3·89-s − 1.09·91-s − 14.3·97-s − 16.0·101-s + 7.76·103-s + ⋯ |
| L(s) = 1 | − 0.158·7-s + 1.96·11-s + 0.728·13-s + 0.551·17-s + 0.980·19-s − 0.891·23-s − 0.487·29-s + 0.538·31-s + 1.00·37-s + 1.36·41-s − 1.58·43-s + 1.53·47-s − 0.974·49-s − 0.762·53-s − 0.450·59-s + 1.31·61-s − 0.744·67-s − 0.311·71-s + 0.761·73-s − 0.310·77-s + 0.705·79-s + 0.0494·83-s − 1.10·89-s − 0.115·91-s − 1.46·97-s − 1.59·101-s + 0.765·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.479697031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.479697031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 - 6.50T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 - 0.450T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
| 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
−8.167179858653295873410162233803, −7.47930164579993423820353745416, −6.60513772852100802140893783701, −6.14703221364194351077388064946, −5.40026579687968520835200509507, −4.27018820696044882666553175493, −3.79363622028468277725570527775, −2.96681069187611341567292406210, −1.68072528637425032066504194584, −0.926102464670511584447071949379,
0.926102464670511584447071949379, 1.68072528637425032066504194584, 2.96681069187611341567292406210, 3.79363622028468277725570527775, 4.27018820696044882666553175493, 5.40026579687968520835200509507, 6.14703221364194351077388064946, 6.60513772852100802140893783701, 7.47930164579993423820353745416, 8.167179858653295873410162233803
