| L(s) = 1 | + 2.67·3-s + 4.29·5-s − 4.01·7-s + 4.16·9-s + 4.88·11-s − 3.92·13-s + 11.5·15-s + 0.345·17-s + 3.26·19-s − 10.7·21-s + 7.48·23-s + 13.4·25-s + 3.12·27-s − 10.5·29-s − 9.70·31-s + 13.0·33-s − 17.2·35-s + 4.92·37-s − 10.5·39-s + 7.39·41-s − 4.90·43-s + 17.9·45-s + 1.34·47-s + 9.14·49-s + 0.925·51-s + 1.36·53-s + 20.9·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 1.92·5-s − 1.51·7-s + 1.38·9-s + 1.47·11-s − 1.08·13-s + 2.97·15-s + 0.0838·17-s + 0.749·19-s − 2.34·21-s + 1.56·23-s + 2.69·25-s + 0.601·27-s − 1.96·29-s − 1.74·31-s + 2.27·33-s − 2.91·35-s + 0.810·37-s − 1.68·39-s + 1.15·41-s − 0.748·43-s + 2.66·45-s + 0.195·47-s + 1.30·49-s + 0.129·51-s + 0.187·53-s + 2.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.893879772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.893879772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 0.345T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 7.39T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 - 1.36T + 53T^{2} \) |
| 59 | \( 1 + 0.330T + 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 2.84T + 97T^{2} \) |
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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
−9.240053453866160732424942671641, −8.980808106847973976148808131251, −7.36223613798542025177822093386, −6.96510968691971887384808950403, −6.05577712669783246099849876179, −5.28293189526810742210223535701, −3.86411333407445473408911486209, −3.07788968656915349979836747614, −2.38282662615593287628788805487, −1.42885448011641612285555482805,
1.42885448011641612285555482805, 2.38282662615593287628788805487, 3.07788968656915349979836747614, 3.86411333407445473408911486209, 5.28293189526810742210223535701, 6.05577712669783246099849876179, 6.96510968691971887384808950403, 7.36223613798542025177822093386, 8.980808106847973976148808131251, 9.240053453866160732424942671641
