L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s + 9-s + 10-s − 5·11-s − 12-s + 13-s + 2·14-s − 15-s + 16-s + 5·17-s + 18-s − 2·19-s + 20-s − 2·21-s − 5·22-s + 4·23-s − 24-s + 25-s + 26-s − 27-s + 2·28-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 − 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.436·21-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.160042767\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.160042767\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.61819926225802, −12.23603865155186, −11.60341641730086, −11.44723221983947, −10.68471441135903, −10.46717164963793, −10.13315511617042, −9.626296404621257, −8.740728689105279, −8.490708667435182, −7.834631121257124, −7.510708064236485, −6.982467383300078, −6.410088314966363, −5.900340353834906, −5.415229417433783, −5.158568452141194, −4.689763601293504, −4.194240154108407, −3.388102208891203, −3.006936293371367, −2.305532846702734, −1.885898811050428, −1.082864887780401, −0.5962644062199701,
0.5962644062199701, 1.082864887780401, 1.885898811050428, 2.305532846702734, 3.006936293371367, 3.388102208891203, 4.194240154108407, 4.689763601293504, 5.158568452141194, 5.415229417433783, 5.900340353834906, 6.410088314966363, 6.982467383300078, 7.510708064236485, 7.834631121257124, 8.490708667435182, 8.740728689105279, 9.626296404621257, 10.13315511617042, 10.46717164963793, 10.68471441135903, 11.44723221983947, 11.60341641730086, 12.23603865155186, 12.61819926225802