| L(s) = 1 | + 3-s − 1.61·7-s − 2·9-s + 0.763·11-s + 4.85·13-s + 0.763·17-s + 5.85·19-s − 1.61·21-s + 8.23·23-s − 5·27-s − 1.38·29-s + 3·31-s + 0.763·33-s − 4.23·37-s + 4.85·39-s − 5.23·41-s + 1.85·43-s − 1.61·47-s − 4.38·49-s + 0.763·51-s − 5.47·53-s + 5.85·57-s + 4.14·59-s − 4.70·61-s + 3.23·63-s + 9.23·67-s + 8.23·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.611·7-s − 0.666·9-s + 0.230·11-s + 1.34·13-s + 0.185·17-s + 1.34·19-s − 0.353·21-s + 1.71·23-s − 0.962·27-s − 0.256·29-s + 0.538·31-s + 0.132·33-s − 0.696·37-s + 0.777·39-s − 0.817·41-s + 0.282·43-s − 0.236·47-s − 0.625·49-s + 0.106·51-s − 0.751·53-s + 0.775·57-s + 0.539·59-s − 0.602·61-s + 0.407·63-s + 1.12·67-s + 0.991·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559855090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559855090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
| 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
−7.79778770897930776602864174938, −6.87202453758690942949578510776, −6.43685246553269959503115364834, −5.55425730382800451091725002234, −5.06208082483664266239202691544, −3.89571684871458772528874407645, −3.25552919703904723880773997353, −2.92422778911542010400748228805, −1.68604461042899986509492295716, −0.75771030409936258827652457902,
0.75771030409936258827652457902, 1.68604461042899986509492295716, 2.92422778911542010400748228805, 3.25552919703904723880773997353, 3.89571684871458772528874407645, 5.06208082483664266239202691544, 5.55425730382800451091725002234, 6.43685246553269959503115364834, 6.87202453758690942949578510776, 7.79778770897930776602864174938
