| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 3·13-s + 4·14-s + 15-s + 16-s − 18-s − 3·19-s + 20-s − 4·21-s + 22-s + 23-s − 24-s + 25-s − 3·26-s + 27-s − 4·28-s − 5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.832·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.872·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s − 0.755·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.94820856716219, −13.49139533504229, −13.01286276961172, −12.78256397029544, −12.17431731801793, −11.49824636012563, −10.91727513222157, −10.40322697534743, −10.06457530590531, −9.502470311437996, −9.115764581305265, −8.699741497649825, −8.183945899471790, −7.449371015126044, −7.128565123849829, −6.331966149078899, −6.146950565460417, −5.608816564488383, −4.662723929993632, −3.989393986959432, −3.419332509895997, −2.835600049390037, −2.398874846148842, −1.611481074534490, −0.8465946270634696, 0,
0.8465946270634696, 1.611481074534490, 2.398874846148842, 2.835600049390037, 3.419332509895997, 3.989393986959432, 4.662723929993632, 5.608816564488383, 6.146950565460417, 6.331966149078899, 7.128565123849829, 7.449371015126044, 8.183945899471790, 8.699741497649825, 9.115764581305265, 9.502470311437996, 10.06457530590531, 10.40322697534743, 10.91727513222157, 11.49824636012563, 12.17431731801793, 12.78256397029544, 13.01286276961172, 13.49139533504229, 13.94820856716219
