| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 5·11-s + 12-s + 13-s + 14-s + 16-s − 18-s + 19-s − 21-s + 5·22-s − 3·23-s − 24-s − 26-s + 27-s − 28-s + 3·31-s − 32-s − 5·33-s + 36-s + 8·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.218·21-s + 1.06·22-s − 0.625·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.538·31-s − 0.176·32-s − 0.870·33-s + 1/6·36-s + 1.31·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.87697039511319, −15.71865991216679, −14.82242927357352, −14.48581798222887, −13.63153745892469, −13.25752211339349, −12.68242422628180, −12.18831869177202, −11.30231530828550, −10.90140378984489, −10.28193709227915, −9.649095679076006, −9.434757726568291, −8.524636119077264, −7.944616482618379, −7.796788011754079, −6.951163308074791, −6.266854084844148, −5.685688152841220, −4.876001212793949, −4.128727127884934, −3.221167458686992, −2.700858922317560, −2.057835084036137, −1.021845019104595, 0,
1.021845019104595, 2.057835084036137, 2.700858922317560, 3.221167458686992, 4.128727127884934, 4.876001212793949, 5.685688152841220, 6.266854084844148, 6.951163308074791, 7.796788011754079, 7.944616482618379, 8.524636119077264, 9.434757726568291, 9.649095679076006, 10.28193709227915, 10.90140378984489, 11.30231530828550, 12.18831869177202, 12.68242422628180, 13.25752211339349, 13.63153745892469, 14.48581798222887, 14.82242927357352, 15.71865991216679, 15.87697039511319
