| L(s) = 1 | + 1.67·3-s + 1.77·5-s + 3.87·7-s − 0.186·9-s + 2.03·11-s − 1.03·13-s + 2.97·15-s + 4.37·17-s − 4.52·19-s + 6.49·21-s + 7.97·23-s − 1.85·25-s − 5.34·27-s + 3.14·29-s − 6.36·31-s + 3.42·33-s + 6.86·35-s + 3.25·37-s − 1.74·39-s − 0.174·41-s + 0.0316·43-s − 0.331·45-s + 8.53·47-s + 8.00·49-s + 7.33·51-s + 9.52·53-s + 3.61·55-s + ⋯ |
| L(s) = 1 | + 0.968·3-s + 0.793·5-s + 1.46·7-s − 0.0622·9-s + 0.614·11-s − 0.288·13-s + 0.768·15-s + 1.06·17-s − 1.03·19-s + 1.41·21-s + 1.66·23-s − 0.370·25-s − 1.02·27-s + 0.583·29-s − 1.14·31-s + 0.595·33-s + 1.16·35-s + 0.535·37-s − 0.279·39-s − 0.0272·41-s + 0.00482·43-s − 0.0494·45-s + 1.24·47-s + 1.14·49-s + 1.02·51-s + 1.30·53-s + 0.487·55-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
| \(L(1)\) |
\(\approx\) |
\(4.476907860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.476907860\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 + 6.36T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 + 0.174T + 41T^{2} \) |
| 43 | \( 1 - 0.0316T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 0.415T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 + 0.0579T + 71T^{2} \) |
| 73 | \( 1 - 6.40T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 0.482T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.83012977591368333936806154516, −7.41585762516515852677417542233, −6.45260465050892495896132385068, −5.63718331765479188997916452026, −5.07799234836121450933248449405, −4.25170508576972484623435447077, −3.43496131257768179997791282868, −2.50288003038654462974871134359, −1.91297503381265311700155691606, −1.07556479230360626025057832723,
1.07556479230360626025057832723, 1.91297503381265311700155691606, 2.50288003038654462974871134359, 3.43496131257768179997791282868, 4.25170508576972484623435447077, 5.07799234836121450933248449405, 5.63718331765479188997916452026, 6.45260465050892495896132385068, 7.41585762516515852677417542233, 7.83012977591368333936806154516
