| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5.23·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 10.4·10-s − 33.6·11-s − 12·12-s − 12.8·13-s − 14·14-s + 15.7·15-s + 16·16-s − 88.6·17-s + 18·18-s + 155.·19-s − 20.9·20-s + 21·21-s − 67.3·22-s + 23·23-s − 24·24-s − 97.5·25-s − 25.7·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.468·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.331·10-s − 0.923·11-s − 0.288·12-s − 0.274·13-s − 0.267·14-s + 0.270·15-s + 0.250·16-s − 1.26·17-s + 0.235·18-s + 1.87·19-s − 0.234·20-s + 0.218·21-s − 0.653·22-s + 0.208·23-s − 0.204·24-s − 0.780·25-s − 0.194·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.895583730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895583730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 5.23T + 125T^{2} \) |
| 11 | \( 1 + 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 305.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 145.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 356.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 626.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 838.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 455.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 656.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 946.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 112.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 192.T + 9.12e5T^{2} \) |
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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
−9.845410478004866580131697535637, −8.798788839051371641043184236642, −7.59564532081245638754083860529, −7.10679559407262044054306576128, −6.03628103470322826775316772207, −5.24148596758355644033183391903, −4.44529777523988519245942298475, −3.37377178549607119124554749200, −2.33147504786790676832111000870, −0.66589705959884805255209418333,
0.66589705959884805255209418333, 2.33147504786790676832111000870, 3.37377178549607119124554749200, 4.44529777523988519245942298475, 5.24148596758355644033183391903, 6.03628103470322826775316772207, 7.10679559407262044054306576128, 7.59564532081245638754083860529, 8.798788839051371641043184236642, 9.845410478004866580131697535637
