| L(s) = 1 | + 3·3-s − 4·7-s + 9·9-s + 28·11-s − 16·13-s + 108·17-s − 32·19-s − 12·21-s + 28·23-s + 27·27-s − 238·29-s + 180·31-s + 84·33-s − 40·37-s − 48·39-s + 422·41-s − 276·43-s − 60·47-s − 327·49-s + 324·51-s + 220·53-s − 96·57-s + 804·59-s − 358·61-s − 36·63-s + 884·67-s + 84·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.215·7-s + 1/3·9-s + 0.767·11-s − 0.341·13-s + 1.54·17-s − 0.386·19-s − 0.124·21-s + 0.253·23-s + 0.192·27-s − 1.52·29-s + 1.04·31-s + 0.443·33-s − 0.177·37-s − 0.197·39-s + 1.60·41-s − 0.978·43-s − 0.186·47-s − 0.953·49-s + 0.889·51-s + 0.570·53-s − 0.223·57-s + 1.77·59-s − 0.751·61-s − 0.0719·63-s + 1.61·67-s + 0.146·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.821082240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821082240\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 108 T + p^{3} T^{2} \) |
| 19 | \( 1 + 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 29 | \( 1 + 238 T + p^{3} T^{2} \) |
| 31 | \( 1 - 180 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 276 T + p^{3} T^{2} \) |
| 47 | \( 1 + 60 T + p^{3} T^{2} \) |
| 53 | \( 1 - 220 T + p^{3} T^{2} \) |
| 59 | \( 1 - 804 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 64 T + p^{3} T^{2} \) |
| 73 | \( 1 + 152 T + p^{3} T^{2} \) |
| 79 | \( 1 - 932 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1292 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 - 824 T + p^{3} T^{2} \) |
| 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
−9.518812700556278241434190629091, −8.537492272757696779113350365224, −7.78697283281724805690538542779, −6.99348074998020086628952366416, −6.07515562748226043974315505830, −5.09216876320199493630293686433, −3.96379811272187602951781959633, −3.22577853474776783255734760040, −2.05679956545380908430919433046, −0.860590112800329172684201059180,
0.860590112800329172684201059180, 2.05679956545380908430919433046, 3.22577853474776783255734760040, 3.96379811272187602951781959633, 5.09216876320199493630293686433, 6.07515562748226043974315505830, 6.99348074998020086628952366416, 7.78697283281724805690538542779, 8.537492272757696779113350365224, 9.518812700556278241434190629091
