| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s + 9-s + 10-s − 2·11-s − 12-s + 2·13-s + 15-s − 16-s + 4·17-s + 18-s − 20-s − 2·22-s − 3·24-s + 25-s + 2·26-s + 27-s + 29-s + 30-s − 2·31-s + 5·32-s − 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.426·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.185·29-s + 0.182·30-s − 0.359·31-s + 0.883·32-s − 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.25834805823177, −12.71284307892473, −12.47510104260186, −12.02921067752997, −11.28041822201049, −10.87164243469175, −10.31254127362104, −9.707626662367716, −9.578443340239439, −8.879711623573686, −8.523318627031309, −8.031967440198849, −7.546808943524987, −6.958354668125705, −6.319804430532592, −5.813705430962008, −5.448983320420372, −4.946711726297277, −4.355152465263521, −3.832346581314016, −3.374726798597497, −2.794540417108486, −2.381283444645239, −1.495094979714609, −0.9228774852725354, 0,
0.9228774852725354, 1.495094979714609, 2.381283444645239, 2.794540417108486, 3.374726798597497, 3.832346581314016, 4.355152465263521, 4.946711726297277, 5.448983320420372, 5.813705430962008, 6.319804430532592, 6.958354668125705, 7.546808943524987, 8.031967440198849, 8.523318627031309, 8.879711623573686, 9.578443340239439, 9.707626662367716, 10.31254127362104, 10.87164243469175, 11.28041822201049, 12.02921067752997, 12.47510104260186, 12.71284307892473, 13.25834805823177
