| L(s) = 1 | + 0.867·3-s − 3.26·5-s − 1.33·7-s − 2.24·9-s − 2.07·11-s + 1.16·13-s − 2.82·15-s + 7.33·17-s + 2.19·19-s − 1.15·21-s − 2.40·23-s + 5.64·25-s − 4.55·27-s + 7.29·29-s + 2.69·31-s − 1.80·33-s + 4.33·35-s + 6.11·37-s + 1.01·39-s − 3.25·41-s − 8.67·43-s + 7.33·45-s − 5.96·47-s − 5.23·49-s + 6.35·51-s + 2.22·53-s + 6.77·55-s + ⋯ |
| L(s) = 1 | + 0.500·3-s − 1.45·5-s − 0.502·7-s − 0.749·9-s − 0.625·11-s + 0.323·13-s − 0.730·15-s + 1.77·17-s + 0.503·19-s − 0.251·21-s − 0.501·23-s + 1.12·25-s − 0.876·27-s + 1.35·29-s + 0.484·31-s − 0.313·33-s + 0.733·35-s + 1.00·37-s + 0.162·39-s − 0.508·41-s − 1.32·43-s + 1.09·45-s − 0.870·47-s − 0.747·49-s + 0.890·51-s + 0.305·53-s + 0.912·55-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 | \( 1 - 0.867T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 - 7.33T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 - 0.844T + 59T^{2} \) |
| 61 | \( 1 - 1.92T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 0.0579T + 73T^{2} \) |
| 79 | \( 1 - 0.496T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.78218272586471489394009714197, −6.92378383152984172389255550326, −6.14112672771408782925786822085, −5.33232254727735887522798417403, −4.61122165134693226909339663728, −3.48864242117222165103454144457, −3.37793833244703707250453492138, −2.54132689184573335576675520980, −1.06603667989526294435090719764, 0,
1.06603667989526294435090719764, 2.54132689184573335576675520980, 3.37793833244703707250453492138, 3.48864242117222165103454144457, 4.61122165134693226909339663728, 5.33232254727735887522798417403, 6.14112672771408782925786822085, 6.92378383152984172389255550326, 7.78218272586471489394009714197
