| L(s) = 1 | + 5-s + 3·7-s + 3·11-s + 4·17-s − 6·19-s + 6·23-s − 4·25-s + 2·29-s + 9·31-s + 3·35-s + 2·37-s − 10·41-s − 6·43-s + 6·47-s + 2·49-s − 13·53-s + 3·55-s + 12·59-s − 8·61-s − 6·67-s + 12·71-s + 9·73-s + 9·77-s + 3·83-s + 4·85-s + 14·89-s − 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.904·11-s + 0.970·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s + 0.371·29-s + 1.61·31-s + 0.507·35-s + 0.328·37-s − 1.56·41-s − 0.914·43-s + 0.875·47-s + 2/7·49-s − 1.78·53-s + 0.404·55-s + 1.56·59-s − 1.02·61-s − 0.733·67-s + 1.42·71-s + 1.05·73-s + 1.02·77-s + 0.329·83-s + 0.433·85-s + 1.48·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.324480606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.324480606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
| 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
−9.304920511567033460814926393629, −8.431694855782295595626508228102, −7.947702036273775290970098315180, −6.80872874731335356584175001406, −6.18749838061342877706828174007, −5.12498950972641249520080852035, −4.48725611586962911066078416540, −3.38459661044143839471648596100, −2.10796110574577917516614712754, −1.16989801505626396377706200589,
1.16989801505626396377706200589, 2.10796110574577917516614712754, 3.38459661044143839471648596100, 4.48725611586962911066078416540, 5.12498950972641249520080852035, 6.18749838061342877706828174007, 6.80872874731335356584175001406, 7.947702036273775290970098315180, 8.431694855782295595626508228102, 9.304920511567033460814926393629
