| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 2·13-s + 14-s + 16-s + 2·19-s + 20-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s + 32-s + 35-s − 4·37-s + 2·38-s + 40-s − 6·41-s + 2·43-s + 6·47-s + 49-s + 50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.169·35-s − 0.657·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582299914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582299914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.72404098936852919349512142802, −9.859293451209498331903316737933, −8.837533638096091066090521817753, −7.87266301798245480225791296096, −6.87293396816033774027000812050, −5.93829815350757951860423599864, −5.12156168153122389070395792926, −4.05964914153577511754110235204, −2.88888582302515315869877464652, −1.54625076555069290237365702205,
1.54625076555069290237365702205, 2.88888582302515315869877464652, 4.05964914153577511754110235204, 5.12156168153122389070395792926, 5.93829815350757951860423599864, 6.87293396816033774027000812050, 7.87266301798245480225791296096, 8.837533638096091066090521817753, 9.859293451209498331903316737933, 10.72404098936852919349512142802
