| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 3·7-s + 8-s + 9-s − 3·10-s − 11-s + 12-s − 3·14-s − 3·15-s + 16-s − 17-s + 18-s + 2·19-s − 3·20-s − 3·21-s − 22-s − 3·23-s + 24-s + 4·25-s + 27-s − 3·28-s − 5·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s − 0.801·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.670·20-s − 0.654·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.566·28-s − 0.928·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.55000429067250, −13.04575451193387, −12.78775445471425, −12.13730407338617, −11.84922903626637, −11.47158800608398, −10.65327209963805, −10.49017675946275, −9.776212003207614, −9.362179773571422, −8.653451412445906, −8.417399940183064, −7.646359907803323, −7.324146041563867, −6.978363302269996, −6.393804739189334, −5.731477054466765, −5.244718369852889, −4.635648027576091, −3.892539231194760, −3.702701589822307, −3.285934296688509, −2.700324771414334, −2.016547286829686, −1.295483985948520, 0, 0,
1.295483985948520, 2.016547286829686, 2.700324771414334, 3.285934296688509, 3.702701589822307, 3.892539231194760, 4.635648027576091, 5.244718369852889, 5.731477054466765, 6.393804739189334, 6.978363302269996, 7.324146041563867, 7.646359907803323, 8.417399940183064, 8.653451412445906, 9.362179773571422, 9.776212003207614, 10.49017675946275, 10.65327209963805, 11.47158800608398, 11.84922903626637, 12.13730407338617, 12.78775445471425, 13.04575451193387, 13.55000429067250
