| L(s) = 1 | − 0.385·3-s − 1.81·5-s − 4.21·7-s − 2.85·9-s + 3.30·11-s + 1.65·13-s + 0.700·15-s + 0.132·17-s − 4.58·19-s + 1.62·21-s − 1.01·23-s − 1.70·25-s + 2.25·27-s + 3.91·29-s − 5.63·31-s − 1.27·33-s + 7.64·35-s − 11.5·37-s − 0.639·39-s − 12.1·41-s − 2.00·43-s + 5.17·45-s − 1.27·47-s + 10.7·49-s − 0.0511·51-s − 2.36·53-s − 6.00·55-s + ⋯ |
| L(s) = 1 | − 0.222·3-s − 0.811·5-s − 1.59·7-s − 0.950·9-s + 0.997·11-s + 0.459·13-s + 0.180·15-s + 0.0321·17-s − 1.05·19-s + 0.354·21-s − 0.212·23-s − 0.341·25-s + 0.434·27-s + 0.726·29-s − 1.01·31-s − 0.222·33-s + 1.29·35-s − 1.89·37-s − 0.102·39-s − 1.89·41-s − 0.306·43-s + 0.771·45-s − 0.185·47-s + 1.53·49-s − 0.00716·51-s − 0.324·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4218523522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4218523522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 0.385T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 - 0.132T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 2.36T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 + 5.95T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.357678905717090885342637346714, −7.10793872846833468819857035116, −6.67456817213318997629758368110, −6.09305144752380176831812623601, −5.34120629195397459715114233032, −4.22336286861924612641387640538, −3.54723243243454608781549091329, −3.11921389602264250045856053033, −1.83656589541445196263883041525, −0.32946340319475784673362183954,
0.32946340319475784673362183954, 1.83656589541445196263883041525, 3.11921389602264250045856053033, 3.54723243243454608781549091329, 4.22336286861924612641387640538, 5.34120629195397459715114233032, 6.09305144752380176831812623601, 6.67456817213318997629758368110, 7.10793872846833468819857035116, 8.357678905717090885342637346714
