| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s − 5·11-s − 12-s + 14-s − 15-s + 16-s + 2·17-s + 2·18-s + 20-s + 21-s + 5·22-s − 2·23-s + 24-s + 25-s + 5·27-s − 28-s − 10·29-s + 30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s + 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.962·27-s − 0.188·28-s − 1.85·29-s + 0.182·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.67333979614488, −15.36761797406576, −14.43861715353459, −14.19778948736624, −13.34943132625556, −12.68963081845339, −12.60889465200684, −11.64409832787729, −11.06459376953319, −10.87439293618358, −10.08890360871256, −9.716247814832219, −9.107530416492763, −8.470645892006419, −7.768529854189450, −7.473116850013234, −6.611181550289489, −5.942941657610738, −5.564752661060543, −5.102943948560559, −4.059970638978121, −3.181616628783630, −2.560014627582439, −1.928259692537141, −0.7660485539230435, 0,
0.7660485539230435, 1.928259692537141, 2.560014627582439, 3.181616628783630, 4.059970638978121, 5.102943948560559, 5.564752661060543, 5.942941657610738, 6.611181550289489, 7.473116850013234, 7.768529854189450, 8.470645892006419, 9.107530416492763, 9.716247814832219, 10.08890360871256, 10.87439293618358, 11.06459376953319, 11.64409832787729, 12.60889465200684, 12.68963081845339, 13.34943132625556, 14.19778948736624, 14.43861715353459, 15.36761797406576, 15.67333979614488
