| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 2·9-s − 10-s − 6·11-s + 12-s + 7·13-s + 15-s + 16-s + 2·18-s − 5·19-s + 20-s + 6·22-s − 6·23-s − 24-s + 25-s − 7·26-s − 5·27-s − 3·29-s − 30-s + 5·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s + 1.94·13-s + 0.258·15-s + 1/4·16-s + 0.471·18-s − 1.14·19-s + 0.223·20-s + 1.27·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.962·27-s − 0.557·29-s − 0.182·30-s + 0.898·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.68783419429950, −13.11251051979404, −12.95039172700706, −12.15546238028343, −11.56752538723511, −11.10670455462215, −10.59926304156425, −10.33197884110834, −9.860969809977607, −9.086324128551756, −8.718318986420356, −8.381666034393911, −7.905566754617159, −7.602094832500245, −6.706951877702120, −6.130513987273120, −5.887073713247755, −5.344721523071705, −4.556906963042450, −3.865991230230814, −3.244036198238913, −2.787393745576077, −2.056724685141385, −1.806001207005768, −0.7359153021848908, 0,
0.7359153021848908, 1.806001207005768, 2.056724685141385, 2.787393745576077, 3.244036198238913, 3.865991230230814, 4.556906963042450, 5.344721523071705, 5.887073713247755, 6.130513987273120, 6.706951877702120, 7.602094832500245, 7.905566754617159, 8.381666034393911, 8.718318986420356, 9.086324128551756, 9.860969809977607, 10.33197884110834, 10.59926304156425, 11.10670455462215, 11.56752538723511, 12.15546238028343, 12.95039172700706, 13.11251051979404, 13.68783419429950
