L(s) = 1 | + 2·7-s − 6·11-s + 2·13-s − 6·17-s − 4·19-s + 8·23-s + 8·31-s − 2·37-s + 6·41-s + 4·43-s + 4·47-s − 3·49-s + 6·53-s − 6·59-s − 6·61-s − 4·71-s − 12·73-s − 12·77-s + 8·79-s − 12·83-s − 14·89-s + 4·91-s − 8·97-s − 8·101-s − 6·103-s − 12·107-s + 2·109-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 1.80·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 1.43·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.781·59-s − 0.768·61-s − 0.474·71-s − 1.40·73-s − 1.36·77-s + 0.900·79-s − 1.31·83-s − 1.48·89-s + 0.419·91-s − 0.812·97-s − 0.796·101-s − 0.591·103-s − 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−7.60733802202789929975567518484, −6.95594491697492412467197053080, −6.15575600439878108091717512223, −5.38502746851780099828685270004, −4.68609987309176058613844090605, −4.21110028709771213075809197193, −2.85086806654383045251024681251, −2.45236351175347052069416208346, −1.29497360122076682923877002647, 0,
1.29497360122076682923877002647, 2.45236351175347052069416208346, 2.85086806654383045251024681251, 4.21110028709771213075809197193, 4.68609987309176058613844090605, 5.38502746851780099828685270004, 6.15575600439878108091717512223, 6.95594491697492412467197053080, 7.60733802202789929975567518484