| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 7-s + 3·8-s + 9-s + 10-s + 2·11-s − 12-s + 14-s − 15-s − 16-s + 17-s − 18-s − 5·19-s + 20-s − 21-s − 2·22-s − 3·23-s + 3·24-s + 25-s + 27-s + 28-s − 5·29-s + 30-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 + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s − 0.625·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.03186517014331, −12.40492986040611, −12.04292285753205, −11.53371805046506, −10.93696050536773, −10.49203491302642, −10.02186516650933, −9.618540823461655, −9.279321854392598, −8.679998917329292, −8.328166097469373, −8.044931508585220, −7.436725678889341, −7.052390149151951, −6.327001507612144, −6.092037037716812, −5.248978840768681, −4.600553537834942, −4.220985791381896, −3.902366142634077, −3.159977502014411, −2.718246870603609, −1.837610262665928, −1.453129702284455, −0.6450638301577262, 0,
0.6450638301577262, 1.453129702284455, 1.837610262665928, 2.718246870603609, 3.159977502014411, 3.902366142634077, 4.220985791381896, 4.600553537834942, 5.248978840768681, 6.092037037716812, 6.327001507612144, 7.052390149151951, 7.436725678889341, 8.044931508585220, 8.328166097469373, 8.679998917329292, 9.279321854392598, 9.618540823461655, 10.02186516650933, 10.49203491302642, 10.93696050536773, 11.53371805046506, 12.04292285753205, 12.40492986040611, 13.03186517014331
