| L(s) = 1 | − 2·5-s + 7-s − 3·9-s − 4·11-s − 2·13-s − 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s + 2·37-s + 2·41-s − 4·43-s + 6·45-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s − 3·63-s + 4·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s − 16·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s − 0.377·63-s + 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.05227520706172484192871939077, −9.688287404638284990728895668207, −8.749304120934004278681759852831, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −3.60173413237973468362473708205, −2.40145156924384381758797773878, 0,
2.40145156924384381758797773878, 3.60173413237973468362473708205, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.749304120934004278681759852831, 9.688287404638284990728895668207, 11.05227520706172484192871939077
