| L(s) = 1 | − 2·7-s − 3·9-s + 2·11-s + 13-s − 6·17-s + 6·19-s + 8·23-s + 2·29-s − 10·31-s + 6·37-s − 6·41-s + 4·43-s − 2·47-s − 3·49-s − 6·53-s + 10·59-s − 2·61-s + 6·63-s + 10·67-s − 10·71-s − 2·73-s − 4·77-s + 4·79-s + 9·81-s − 6·83-s − 6·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 0.603·11-s + 0.277·13-s − 1.45·17-s + 1.37·19-s + 1.66·23-s + 0.371·29-s − 1.79·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s + 1.30·59-s − 0.256·61-s + 0.755·63-s + 1.22·67-s − 1.18·71-s − 0.234·73-s − 0.455·77-s + 0.450·79-s + 81-s − 0.658·83-s − 0.635·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.86747778253058428321634144597, −6.90989103597164361123164904710, −6.58561421425352810799573003367, −5.64200925149600114046948208839, −5.05857580432991311603755392510, −3.99979793257417046798373592279, −3.22375990283750942662629613494, −2.57589324664405212099117480326, −1.27686040667956350994960425186, 0,
1.27686040667956350994960425186, 2.57589324664405212099117480326, 3.22375990283750942662629613494, 3.99979793257417046798373592279, 5.05857580432991311603755392510, 5.64200925149600114046948208839, 6.58561421425352810799573003367, 6.90989103597164361123164904710, 7.86747778253058428321634144597
