| L(s) = 1 | + 0.857·3-s + 5-s + 2.23·7-s − 2.26·9-s − 3.21·11-s + 3.25·13-s + 0.857·15-s + 0.870·17-s + 5.02·19-s + 1.91·21-s + 8.46·23-s + 25-s − 4.51·27-s + 29-s + 0.0467·31-s − 2.75·33-s + 2.23·35-s + 11.1·37-s + 2.78·39-s − 2.47·41-s + 2.87·43-s − 2.26·45-s + 2.20·47-s − 1.99·49-s + 0.746·51-s − 11.7·53-s − 3.21·55-s + ⋯ |
| L(s) = 1 | + 0.495·3-s + 0.447·5-s + 0.845·7-s − 0.754·9-s − 0.970·11-s + 0.901·13-s + 0.221·15-s + 0.211·17-s + 1.15·19-s + 0.418·21-s + 1.76·23-s + 0.200·25-s − 0.869·27-s + 0.185·29-s + 0.00839·31-s − 0.480·33-s + 0.378·35-s + 1.82·37-s + 0.446·39-s − 0.386·41-s + 0.437·43-s − 0.337·45-s + 0.321·47-s − 0.284·49-s + 0.104·51-s − 1.61·53-s − 0.433·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240834519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240834519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.857T + 3T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 0.870T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 31 | \( 1 - 0.0467T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 2.20T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 + 4.99T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 3.43T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 6.44T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−9.573806472521627510606059436679, −9.004091646518504259228457657671, −8.064906799861723339715545867817, −7.64675011836398438511222909282, −6.35618545730569317852916747172, −5.44735526931506380588356651845, −4.79239369429581937718802443433, −3.33150449249403906251612336501, −2.58334215227572763483064408962, −1.21399533910978966136811152060,
1.21399533910978966136811152060, 2.58334215227572763483064408962, 3.33150449249403906251612336501, 4.79239369429581937718802443433, 5.44735526931506380588356651845, 6.35618545730569317852916747172, 7.64675011836398438511222909282, 8.064906799861723339715545867817, 9.004091646518504259228457657671, 9.573806472521627510606059436679
