| L(s) = 1 | − 3·5-s + 7-s − 3·11-s + 4·13-s + 2·19-s + 6·23-s + 4·25-s − 6·29-s − 5·31-s − 3·35-s − 2·37-s − 6·41-s − 10·43-s − 6·47-s − 6·49-s − 9·53-s + 9·55-s + 12·59-s − 8·61-s − 12·65-s + 14·67-s − 7·73-s − 3·77-s − 8·79-s − 3·83-s − 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.904·11-s + 1.10·13-s + 0.458·19-s + 1.25·23-s + 4/5·25-s − 1.11·29-s − 0.898·31-s − 0.507·35-s − 0.328·37-s − 0.937·41-s − 1.52·43-s − 0.875·47-s − 6/7·49-s − 1.23·53-s + 1.21·55-s + 1.56·59-s − 1.02·61-s − 1.48·65-s + 1.71·67-s − 0.819·73-s − 0.341·77-s − 0.900·79-s − 0.329·83-s − 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.646897306767688973776858314960, −8.177655093885067516195802420137, −7.44444168021687832363479010767, −6.72818725632924944067815958116, −5.48544965844839209036398212504, −4.81849202265609017039962796294, −3.72379428045727412120464814901, −3.13977056925804084341234249361, −1.55100613792804333036729509228, 0,
1.55100613792804333036729509228, 3.13977056925804084341234249361, 3.72379428045727412120464814901, 4.81849202265609017039962796294, 5.48544965844839209036398212504, 6.72818725632924944067815958116, 7.44444168021687832363479010767, 8.177655093885067516195802420137, 8.646897306767688973776858314960
