| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 2·7-s − 3·8-s + 9-s + 2·11-s − 12-s + 2·14-s − 16-s + 17-s + 18-s + 2·21-s + 2·22-s − 8·23-s − 3·24-s − 5·25-s + 27-s − 2·28-s − 6·29-s − 6·31-s + 5·32-s + 2·33-s + 34-s − 36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.534·14-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.436·21-s + 0.426·22-s − 1.66·23-s − 0.612·24-s − 25-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.07·31-s + 0.883·32-s + 0.348·33-s + 0.171·34-s − 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.68663893790296892092052717603, −6.62483765835532482862330358509, −5.83936817067815430191112721654, −5.35258431029670249305149165758, −4.40852560362932568992330999290, −3.94815414962252406419987985977, −3.36904283027623159272961987279, −2.24658230100470354947264055635, −1.50508903420130784999279418587, 0,
1.50508903420130784999279418587, 2.24658230100470354947264055635, 3.36904283027623159272961987279, 3.94815414962252406419987985977, 4.40852560362932568992330999290, 5.35258431029670249305149165758, 5.83936817067815430191112721654, 6.62483765835532482862330358509, 7.68663893790296892092052717603
