| L(s) = 1 | + 0.566·2-s + 2.32·3-s − 1.67·4-s + 3.58·5-s + 1.31·6-s − 2.08·8-s + 2.39·9-s + 2.02·10-s + 11-s − 3.90·12-s − 2.11·13-s + 8.32·15-s + 2.17·16-s + 7.87·17-s + 1.35·18-s − 5.56·19-s − 6.01·20-s + 0.566·22-s − 1.25·23-s − 4.83·24-s + 7.83·25-s − 1.19·26-s − 1.40·27-s − 0.991·29-s + 4.71·30-s − 6.06·31-s + 5.39·32-s + ⋯ |
| L(s) = 1 | + 0.400·2-s + 1.34·3-s − 0.839·4-s + 1.60·5-s + 0.536·6-s − 0.736·8-s + 0.798·9-s + 0.641·10-s + 0.301·11-s − 1.12·12-s − 0.586·13-s + 2.14·15-s + 0.544·16-s + 1.91·17-s + 0.319·18-s − 1.27·19-s − 1.34·20-s + 0.120·22-s − 0.261·23-s − 0.987·24-s + 1.56·25-s − 0.234·26-s − 0.270·27-s − 0.184·29-s + 0.859·30-s − 1.09·31-s + 0.954·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.730281576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.730281576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.566T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 - 7.87T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 0.991T + 29T^{2} \) |
| 31 | \( 1 + 6.06T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 + 5.35T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 0.481T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 - 3.58T + 89T^{2} \) |
| 97 | \( 1 - 0.164T + 97T^{2} \) |
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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
−10.21669301023926917223250888930, −9.902486215713594192926504408952, −9.004081035090090135682489450646, −8.505455282089661292763524243899, −7.32184528555123041271931926005, −5.95522124784685164641838436883, −5.26873167160457150233481727073, −3.93302146905185047183585991821, −2.92059387744415445777463427238, −1.77516287553698706359422148543,
1.77516287553698706359422148543, 2.92059387744415445777463427238, 3.93302146905185047183585991821, 5.26873167160457150233481727073, 5.95522124784685164641838436883, 7.32184528555123041271931926005, 8.505455282089661292763524243899, 9.004081035090090135682489450646, 9.902486215713594192926504408952, 10.21669301023926917223250888930
