| L(s) = 1 | − 2-s − 3·3-s + 4-s − 2·5-s + 3·6-s + 7-s − 8-s + 6·9-s + 2·10-s + 11-s − 3·12-s + 7·13-s − 14-s + 6·15-s + 16-s − 3·17-s − 6·18-s − 2·20-s − 3·21-s − 22-s + 3·23-s + 3·24-s − 25-s − 7·26-s − 9·27-s + 28-s − 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s + 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 0.632·10-s + 0.301·11-s − 0.866·12-s + 1.94·13-s − 0.267·14-s + 1.54·15-s + 1/4·16-s − 0.727·17-s − 1.41·18-s − 0.447·20-s − 0.654·21-s − 0.213·22-s + 0.625·23-s + 0.612·24-s − 1/5·25-s − 1.37·26-s − 1.73·27-s + 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.56891805032563490356970753775, −6.64377508580220293249150093596, −6.22147499388149241476252316793, −5.65359920091342225333113667452, −4.62954857802467523299449998726, −4.14162415873467425133031830764, −3.22635364943880482649332003324, −1.67129328262541880123849899590, −0.988575506729953049874948003886, 0,
0.988575506729953049874948003886, 1.67129328262541880123849899590, 3.22635364943880482649332003324, 4.14162415873467425133031830764, 4.62954857802467523299449998726, 5.65359920091342225333113667452, 6.22147499388149241476252316793, 6.64377508580220293249150093596, 7.56891805032563490356970753775
