| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 2·14-s − 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 2·21-s + 22-s − 9·23-s − 24-s + 25-s − 27-s − 2·28-s − 7·29-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 − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s − 1.87·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.29·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.399439381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399439381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.98308347360848, −13.24201360945069, −12.89510975776307, −12.64316280568526, −11.87689250332982, −11.59285924257770, −11.04768535163446, −10.53517826597706, −9.835546425177281, −9.549530359493247, −9.213953401840539, −8.119481326340748, −7.782486000017567, −7.130114884120336, −6.586792938642917, −6.106146073143453, −5.614047431250583, −5.338317580585266, −4.470886083539627, −3.893758167756731, −3.497786997325134, −2.723560079556356, −2.016774227009904, −1.418870470305937, −0.4485969371312215,
0.4485969371312215, 1.418870470305937, 2.016774227009904, 2.723560079556356, 3.497786997325134, 3.893758167756731, 4.470886083539627, 5.338317580585266, 5.614047431250583, 6.106146073143453, 6.586792938642917, 7.130114884120336, 7.782486000017567, 8.119481326340748, 9.213953401840539, 9.549530359493247, 9.835546425177281, 10.53517826597706, 11.04768535163446, 11.59285924257770, 11.87689250332982, 12.64316280568526, 12.89510975776307, 13.24201360945069, 13.98308347360848
