| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 3·11-s − 4·13-s − 4·14-s + 16-s − 3·17-s + 5·19-s − 20-s − 3·22-s − 6·23-s + 25-s − 4·26-s − 4·28-s − 6·29-s + 2·31-s + 32-s − 3·34-s + 4·35-s − 4·37-s + 5·38-s − 40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s − 0.223·20-s − 0.639·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s + 0.676·35-s − 0.657·37-s + 0.811·38-s − 0.158·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.884661439723587617400500749459, −9.181012604635816139449497717962, −7.77404691036140567875834676085, −7.23347773420758791578828913025, −6.22301446970872085611681593876, −5.38292631066779946248479008415, −4.29111105534765483466139869890, −3.28121157426783676388528803494, −2.41912201488827531235600451311, 0,
2.41912201488827531235600451311, 3.28121157426783676388528803494, 4.29111105534765483466139869890, 5.38292631066779946248479008415, 6.22301446970872085611681593876, 7.23347773420758791578828913025, 7.77404691036140567875834676085, 9.181012604635816139449497717962, 9.884661439723587617400500749459
