| L(s) = 1 | − 2·7-s − 11-s − 2·13-s + 4·17-s − 6·19-s − 8·29-s + 8·31-s + 10·37-s − 8·41-s + 2·43-s + 8·47-s − 3·49-s + 2·53-s − 12·59-s − 10·61-s − 12·67-s + 8·71-s − 6·73-s + 2·77-s + 2·79-s + 16·83-s + 14·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 1.48·29-s + 1.43·31-s + 1.64·37-s − 1.24·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s − 1.28·61-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.227·77-s + 0.225·79-s + 1.75·83-s + 1.48·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.47350253754596, −13.07288031660504, −12.51481072212811, −12.21691998926083, −11.73280143289487, −11.09393806247876, −10.55815210657735, −10.23150083180377, −9.725632381955419, −9.186705466431305, −8.895873139562659, −7.978747088594795, −7.826476387483468, −7.279849343719170, −6.538602295964479, −6.192854914035897, −5.773550013640517, −5.043152082007297, −4.542492816842399, −4.008951617451946, −3.327245833598506, −2.856803073074107, −2.262327452331401, −1.593029981166796, −0.6977870324441098, 0,
0.6977870324441098, 1.593029981166796, 2.262327452331401, 2.856803073074107, 3.327245833598506, 4.008951617451946, 4.542492816842399, 5.043152082007297, 5.773550013640517, 6.192854914035897, 6.538602295964479, 7.279849343719170, 7.826476387483468, 7.978747088594795, 8.895873139562659, 9.186705466431305, 9.725632381955419, 10.23150083180377, 10.55815210657735, 11.09393806247876, 11.73280143289487, 12.21691998926083, 12.51481072212811, 13.07288031660504, 13.47350253754596
