| L(s) = 1 | − 2-s + 2.81·3-s + 4-s + 3.87·5-s − 2.81·6-s − 2.87·7-s − 8-s + 4.94·9-s − 3.87·10-s − 3.63·11-s + 2.81·12-s + 2.87·14-s + 10.9·15-s + 16-s + 6.24·17-s − 4.94·18-s − 19-s + 3.87·20-s − 8.08·21-s + 3.63·22-s − 8.25·23-s − 2.81·24-s + 10.0·25-s + 5.47·27-s − 2.87·28-s + 4.57·29-s − 10.9·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.62·3-s + 0.5·4-s + 1.73·5-s − 1.15·6-s − 1.08·7-s − 0.353·8-s + 1.64·9-s − 1.22·10-s − 1.09·11-s + 0.813·12-s + 0.767·14-s + 2.82·15-s + 0.250·16-s + 1.51·17-s − 1.16·18-s − 0.229·19-s + 0.867·20-s − 1.76·21-s + 0.775·22-s − 1.72·23-s − 0.575·24-s + 2.00·25-s + 1.05·27-s − 0.542·28-s + 0.849·29-s − 1.99·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.387350698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.387350698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 23 | \( 1 + 8.25T + 23T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 + 0.0117T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 + 0.214T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 6.62T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6.33T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 + 7.06T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−8.091455415316325910245353228510, −7.60781733342552079036833129105, −6.72768877706808069467053075812, −5.96009844083485875364777708335, −5.49901594213112690141424003534, −4.13290730808014223394413889918, −3.15092098779159469510972622052, −2.54914381526050398881517382250, −2.15247878779817438987660361555, −0.992152768620149376161941001458,
0.992152768620149376161941001458, 2.15247878779817438987660361555, 2.54914381526050398881517382250, 3.15092098779159469510972622052, 4.13290730808014223394413889918, 5.49901594213112690141424003534, 5.96009844083485875364777708335, 6.72768877706808069467053075812, 7.60781733342552079036833129105, 8.091455415316325910245353228510
