| L(s) = 1 | + 3-s − 5·7-s + 9-s − 5·11-s − 13-s + 3·17-s + 4·19-s − 5·21-s + 5·23-s + 27-s − 4·29-s − 5·33-s + 7·37-s − 39-s + 11·41-s − 12·43-s + 6·47-s + 18·49-s + 3·51-s − 53-s + 4·57-s − 12·59-s − 7·61-s − 5·63-s + 4·67-s + 5·69-s + 7·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s − 1.09·21-s + 1.04·23-s + 0.192·27-s − 0.742·29-s − 0.870·33-s + 1.15·37-s − 0.160·39-s + 1.71·41-s − 1.82·43-s + 0.875·47-s + 18/7·49-s + 0.420·51-s − 0.137·53-s + 0.529·57-s − 1.56·59-s − 0.896·61-s − 0.629·63-s + 0.488·67-s + 0.601·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−16.27747465937068, −15.67764678504912, −15.27944972238259, −14.73884648537129, −13.84715220461353, −13.46940490273772, −12.97395238885669, −12.54228115636602, −12.06586644587161, −10.98026471179852, −10.61889933535361, −9.784062137656885, −9.539496662698404, −9.109094819422372, −8.030973058106283, −7.704061279858169, −7.039751776083989, −6.423902606636255, −5.614819459670117, −5.174481956969593, −4.158335548382604, −3.323073478576481, −2.942409103243580, −2.386289832912213, −1.011506691907529, 0,
1.011506691907529, 2.386289832912213, 2.942409103243580, 3.323073478576481, 4.158335548382604, 5.174481956969593, 5.614819459670117, 6.423902606636255, 7.039751776083989, 7.704061279858169, 8.030973058106283, 9.109094819422372, 9.539496662698404, 9.784062137656885, 10.61889933535361, 10.98026471179852, 12.06586644587161, 12.54228115636602, 12.97395238885669, 13.46940490273772, 13.84715220461353, 14.73884648537129, 15.27944972238259, 15.67764678504912, 16.27747465937068
