| L(s) = 1 | − 5-s − 2.46·7-s − 4.19·11-s − 3.26·13-s + 7.73·17-s + 0.732·19-s + 23-s + 25-s − 7.19·29-s − 31-s + 2.46·35-s − 11.3·37-s + 7.73·41-s + 3.46·43-s − 0.732·47-s − 0.928·49-s − 6.66·53-s + 4.19·55-s + 7.19·59-s − 10.7·61-s + 3.26·65-s + 5·67-s + 14.1·71-s − 11.2·73-s + 10.3·77-s + 4·79-s + 6.66·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.931·7-s − 1.26·11-s − 0.906·13-s + 1.87·17-s + 0.167·19-s + 0.208·23-s + 0.200·25-s − 1.33·29-s − 0.179·31-s + 0.416·35-s − 1.87·37-s + 1.20·41-s + 0.528·43-s − 0.106·47-s − 0.132·49-s − 0.914·53-s + 0.565·55-s + 0.936·59-s − 1.37·61-s + 0.405·65-s + 0.610·67-s + 1.67·71-s − 1.31·73-s + 1.17·77-s + 0.450·79-s + 0.731·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9766443016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9766443016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 7.73T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 0.732T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.271666437361429482486957972676, −7.48471910346565190155131549742, −7.31013820306469509939203608450, −6.11824043870575387337461853206, −5.42390838381168569674357040944, −4.80811105984273144911628066603, −3.53442423262008260350546858260, −3.15549718158090759245168180559, −2.07921534633835402326857337660, −0.53604838269124823699051315911,
0.53604838269124823699051315911, 2.07921534633835402326857337660, 3.15549718158090759245168180559, 3.53442423262008260350546858260, 4.80811105984273144911628066603, 5.42390838381168569674357040944, 6.11824043870575387337461853206, 7.31013820306469509939203608450, 7.48471910346565190155131549742, 8.271666437361429482486957972676
