| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4·7-s + 8-s + 9-s + 10-s − 12-s + 13-s − 4·14-s − 15-s + 16-s + 4·17-s + 18-s + 3·19-s + 20-s + 4·21-s + 6·23-s − 24-s + 25-s + 26-s − 27-s − 4·28-s − 5·29-s − 30-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 − 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.688·19-s + 0.223·20-s + 0.872·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.71699199356286, −14.40534584104227, −13.54008582131647, −13.29645075580792, −12.85443734161605, −12.41418290136528, −11.83943026031365, −11.33665020159416, −10.73715893695279, −10.12628230074778, −9.837925635995215, −9.171279532725740, −8.730410725830519, −7.634673026986907, −7.278450552139380, −6.667844422656924, −6.203727870283518, −5.638347198071535, −5.262330046953415, −4.588731678252897, −3.594750021020335, −3.378504731547755, −2.743566240574983, −1.754598616467007, −1.024107031150430, 0,
1.024107031150430, 1.754598616467007, 2.743566240574983, 3.378504731547755, 3.594750021020335, 4.588731678252897, 5.262330046953415, 5.638347198071535, 6.203727870283518, 6.667844422656924, 7.278450552139380, 7.634673026986907, 8.730410725830519, 9.171279532725740, 9.837925635995215, 10.12628230074778, 10.73715893695279, 11.33665020159416, 11.83943026031365, 12.41418290136528, 12.85443734161605, 13.29645075580792, 13.54008582131647, 14.40534584104227, 14.71699199356286
