| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 4·10-s − 5·11-s + 13-s − 4·16-s − 17-s + 6·19-s − 4·20-s + 10·22-s + 4·23-s − 25-s − 2·26-s − 8·29-s − 31-s + 8·32-s + 2·34-s + 37-s − 12·38-s − 3·41-s + 11·43-s − 10·44-s − 8·46-s + 2·47-s + 2·50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s − 1.50·11-s + 0.277·13-s − 16-s − 0.242·17-s + 1.37·19-s − 0.894·20-s + 2.13·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 1.48·29-s − 0.179·31-s + 1.41·32-s + 0.342·34-s + 0.164·37-s − 1.94·38-s − 0.468·41-s + 1.67·43-s − 1.50·44-s − 1.17·46-s + 0.291·47-s + 0.282·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−13.84574171743856, −13.56837789207830, −13.04240933935658, −12.44413518711752, −11.94659389845497, −11.22615799529595, −10.99539572203582, −10.69846996467269, −9.937572336813668, −9.523011132529078, −9.097183858401907, −8.526068294721083, −7.936254094776502, −7.605628833984383, −7.396773830579257, −6.728007219467475, −5.945966974540609, −5.192839197877533, −4.973733076436315, −3.968270823426031, −3.596415572857738, −2.696991108175220, −2.266413487790798, −1.339838119096271, −0.6625246472071963, 0,
0.6625246472071963, 1.339838119096271, 2.266413487790798, 2.696991108175220, 3.596415572857738, 3.968270823426031, 4.973733076436315, 5.192839197877533, 5.945966974540609, 6.728007219467475, 7.396773830579257, 7.605628833984383, 7.936254094776502, 8.526068294721083, 9.097183858401907, 9.523011132529078, 9.937572336813668, 10.69846996467269, 10.99539572203582, 11.22615799529595, 11.94659389845497, 12.44413518711752, 13.04240933935658, 13.56837789207830, 13.84574171743856
