| L(s) = 1 | + 2.34·3-s + 2.39·5-s + 3.54·7-s + 2.50·9-s + 4.15·11-s + 3.77·13-s + 5.62·15-s − 5.89·17-s + 2.10·19-s + 8.32·21-s + 0.956·23-s + 0.741·25-s − 1.16·27-s − 10.1·29-s + 1.81·31-s + 9.73·33-s + 8.50·35-s + 2.63·37-s + 8.85·39-s − 2.88·41-s + 9.00·43-s + 5.99·45-s − 2.83·47-s + 5.59·49-s − 13.8·51-s − 7.26·53-s + 9.94·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s + 1.07·5-s + 1.34·7-s + 0.834·9-s + 1.25·11-s + 1.04·13-s + 1.45·15-s − 1.42·17-s + 0.483·19-s + 1.81·21-s + 0.199·23-s + 0.148·25-s − 0.224·27-s − 1.87·29-s + 0.326·31-s + 1.69·33-s + 1.43·35-s + 0.433·37-s + 1.41·39-s − 0.450·41-s + 1.37·43-s + 0.893·45-s − 0.413·47-s + 0.798·49-s − 1.93·51-s − 0.997·53-s + 1.34·55-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
| \(L(1)\) |
\(\approx\) |
\(5.368744774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.368744774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 0.956T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 0.157T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−8.253634940518521836188122133797, −7.54785348936864403667045411077, −6.70379753990914122018578169211, −5.98749365323460614498608291282, −5.20253228390668006084423654025, −4.19581883673472896204984518849, −3.72105258416426060368616365947, −2.57826244612318890835631423731, −1.84611519784816886674064299699, −1.36334794457252965977863678241,
1.36334794457252965977863678241, 1.84611519784816886674064299699, 2.57826244612318890835631423731, 3.72105258416426060368616365947, 4.19581883673472896204984518849, 5.20253228390668006084423654025, 5.98749365323460614498608291282, 6.70379753990914122018578169211, 7.54785348936864403667045411077, 8.253634940518521836188122133797
