| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 14.4·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 28.9·10-s − 21.4·11-s + 12·12-s + 13·13-s + 14·14-s + 43.4·15-s + 16·16-s − 9.52·17-s − 18·18-s + 59.2·19-s + 57.8·20-s − 21·21-s + 42.8·22-s + 52.4·23-s − 24·24-s + 84.4·25-s − 26·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.29·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.915·10-s − 0.586·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.747·15-s + 0.250·16-s − 0.135·17-s − 0.235·18-s + 0.715·19-s + 0.647·20-s − 0.218·21-s + 0.415·22-s + 0.475·23-s − 0.204·24-s + 0.675·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.226588107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.226588107\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 14.4T + 125T^{2} \) |
| 11 | \( 1 + 21.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 9.52T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 187.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 47.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 97.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 415.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 236.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 616.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 804.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 426.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 53.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 231.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 544.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 757.T + 9.12e5T^{2} \) |
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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
−10.07604587570327575790661242359, −9.610246187058194191315401299209, −8.744725625392137467745407792799, −7.907953302906805534844621739498, −6.78462826264030740305049553691, −6.01936692331238868047561725943, −4.86653453925825502322449348609, −3.14719554725447090063504253799, −2.29541658489760747987674821940, −1.02812276399240854191916022978,
1.02812276399240854191916022978, 2.29541658489760747987674821940, 3.14719554725447090063504253799, 4.86653453925825502322449348609, 6.01936692331238868047561725943, 6.78462826264030740305049553691, 7.907953302906805534844621739498, 8.744725625392137467745407792799, 9.610246187058194191315401299209, 10.07604587570327575790661242359
