| L(s) = 1 | + 2·7-s − 4·11-s − 13-s + 17-s − 2·19-s + 7·23-s + 6·29-s − 4·31-s − 11·37-s + 9·41-s + 6·43-s − 4·47-s − 3·49-s − 53-s − 59-s + 5·61-s − 2·67-s + 3·71-s − 4·73-s − 8·77-s + 6·79-s − 11·83-s − 2·91-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 0.277·13-s + 0.242·17-s − 0.458·19-s + 1.45·23-s + 1.11·29-s − 0.718·31-s − 1.80·37-s + 1.40·41-s + 0.914·43-s − 0.583·47-s − 3/7·49-s − 0.137·53-s − 0.130·59-s + 0.640·61-s − 0.244·67-s + 0.356·71-s − 0.468·73-s − 0.911·77-s + 0.675·79-s − 1.20·83-s − 0.209·91-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.755474834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.755474834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.40965793337860, −12.11951670981318, −11.36926775371292, −11.08526574032799, −10.65600143621498, −10.33210718610072, −9.777485877366698, −9.231548026673357, −8.716425975533835, −8.417563671210086, −7.754902185237856, −7.589807988744346, −6.918999826777332, −6.594129279339611, −5.808031921567359, −5.389668368663424, −4.970573988546597, −4.632386689783814, −3.977202174239158, −3.363581268355564, −2.707348785021781, −2.461618398198531, −1.653833044169292, −1.150764526131238, −0.3475109655564929,
0.3475109655564929, 1.150764526131238, 1.653833044169292, 2.461618398198531, 2.707348785021781, 3.363581268355564, 3.977202174239158, 4.632386689783814, 4.970573988546597, 5.389668368663424, 5.808031921567359, 6.594129279339611, 6.918999826777332, 7.589807988744346, 7.754902185237856, 8.417563671210086, 8.716425975533835, 9.231548026673357, 9.777485877366698, 10.33210718610072, 10.65600143621498, 11.08526574032799, 11.36926775371292, 12.11951670981318, 12.40965793337860
