| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s − 2·9-s + 2·10-s − 2·11-s − 12-s − 2·13-s + 2·15-s + 16-s + 2·18-s + 3·19-s − 2·20-s + 2·22-s + 24-s − 25-s + 2·26-s + 5·27-s − 3·29-s − 2·30-s − 5·31-s − 32-s + 2·33-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 0.471·18-s + 0.688·19-s − 0.447·20-s + 0.426·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s − 0.557·29-s − 0.365·30-s − 0.898·31-s − 0.176·32-s + 0.348·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.91943554755154, −15.37923412530424, −14.71336744709318, −14.38761202199375, −13.58278636338484, −12.88568943472660, −12.40732626055099, −11.74071163232978, −11.44532638805631, −11.07477816133221, −10.23256618355535, −10.01489869544516, −9.014128950584599, −8.791489788342523, −7.906502911403719, −7.580415102744214, −7.132090220036451, −6.247810060059067, −5.748997913064702, −5.085453216888849, −4.513825722097249, −3.510111251098481, −3.054755061203928, −2.198829226001966, −1.266725742298345, 0, 0,
1.266725742298345, 2.198829226001966, 3.054755061203928, 3.510111251098481, 4.513825722097249, 5.085453216888849, 5.748997913064702, 6.247810060059067, 7.132090220036451, 7.580415102744214, 7.906502911403719, 8.791489788342523, 9.014128950584599, 10.01489869544516, 10.23256618355535, 11.07477816133221, 11.44532638805631, 11.74071163232978, 12.40732626055099, 12.88568943472660, 13.58278636338484, 14.38761202199375, 14.71336744709318, 15.37923412530424, 15.91943554755154
