| L(s) = 1 | + 4·5-s + 2·7-s + 11-s + 4·13-s + 2·17-s − 6·23-s + 11·25-s − 10·29-s + 8·31-s + 8·35-s − 2·37-s − 2·41-s − 4·43-s − 2·47-s − 3·49-s − 4·53-s + 4·55-s − 8·61-s + 16·65-s + 12·67-s + 2·71-s − 6·73-s + 2·77-s − 10·79-s + 4·83-s + 8·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s + 0.301·11-s + 1.10·13-s + 0.485·17-s − 1.25·23-s + 11/5·25-s − 1.85·29-s + 1.43·31-s + 1.35·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.549·53-s + 0.539·55-s − 1.02·61-s + 1.98·65-s + 1.46·67-s + 0.237·71-s − 0.702·73-s + 0.227·77-s − 1.12·79-s + 0.439·83-s + 0.867·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.751916538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.751916538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.566821806611680639850077758320, −8.658067696563819741951951018305, −8.002473569473077980144979103163, −6.80851165776554556739631289097, −6.00827641926392441460269833363, −5.54375770586906787734164810731, −4.52618706150717552349846730816, −3.32998016913655743888290215953, −2.02183815494364731591532463402, −1.39701758871920656661437952510,
1.39701758871920656661437952510, 2.02183815494364731591532463402, 3.32998016913655743888290215953, 4.52618706150717552349846730816, 5.54375770586906787734164810731, 6.00827641926392441460269833363, 6.80851165776554556739631289097, 8.002473569473077980144979103163, 8.658067696563819741951951018305, 9.566821806611680639850077758320
