| L(s) = 1 | + 1.12·3-s + 2.61·5-s − 0.504·7-s − 1.72·9-s − 0.941·11-s − 4.19·13-s + 2.94·15-s − 2.87·17-s + 5.76·19-s − 0.568·21-s + 2.15·23-s + 1.83·25-s − 5.33·27-s − 4.90·29-s − 0.527·31-s − 1.06·33-s − 1.31·35-s + 3.99·37-s − 4.73·39-s − 1.62·41-s − 0.186·43-s − 4.51·45-s − 5.70·47-s − 6.74·49-s − 3.23·51-s − 1.93·53-s − 2.46·55-s + ⋯ |
| L(s) = 1 | + 0.651·3-s + 1.16·5-s − 0.190·7-s − 0.575·9-s − 0.283·11-s − 1.16·13-s + 0.761·15-s − 0.696·17-s + 1.32·19-s − 0.124·21-s + 0.450·23-s + 0.367·25-s − 1.02·27-s − 0.909·29-s − 0.0948·31-s − 0.184·33-s − 0.222·35-s + 0.656·37-s − 0.757·39-s − 0.253·41-s − 0.0283·43-s − 0.673·45-s − 0.832·47-s − 0.963·49-s − 0.453·51-s − 0.265·53-s − 0.332·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 0.504T + 7T^{2} \) |
| 11 | \( 1 + 0.941T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 + 0.527T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 + 0.186T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 5.26T + 59T^{2} \) |
| 61 | \( 1 + 4.08T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - 8.10T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−7.61108761262780812373107028358, −6.78730922095668674483438855633, −6.08023960940841861987221972729, −5.32918517011161262244474627952, −4.90502878668258704982564313399, −3.71186769038500386668569180278, −2.86551421792682372342804427598, −2.38710297768358813770621458620, −1.52319930372478794170652181318, 0,
1.52319930372478794170652181318, 2.38710297768358813770621458620, 2.86551421792682372342804427598, 3.71186769038500386668569180278, 4.90502878668258704982564313399, 5.32918517011161262244474627952, 6.08023960940841861987221972729, 6.78730922095668674483438855633, 7.61108761262780812373107028358
