| L(s) = 1 | + 4.23·5-s − 7-s + 2.61·11-s + 3.47·13-s + 5.85·17-s + 19-s + 8.23·23-s + 12.9·25-s + 4.61·29-s − 3.85·31-s − 4.23·35-s − 8.23·37-s − 11.5·41-s + 4.47·43-s − 7.47·47-s + 49-s − 6.09·53-s + 11.0·55-s − 11·59-s − 4.23·61-s + 14.7·65-s − 7.56·67-s + 3.76·71-s + 3.61·73-s − 2.61·77-s − 4.47·79-s − 10.5·83-s + ⋯ |
| L(s) = 1 | + 1.89·5-s − 0.377·7-s + 0.789·11-s + 0.962·13-s + 1.41·17-s + 0.229·19-s + 1.71·23-s + 2.58·25-s + 0.857·29-s − 0.692·31-s − 0.716·35-s − 1.35·37-s − 1.80·41-s + 0.681·43-s − 1.08·47-s + 0.142·49-s − 0.836·53-s + 1.49·55-s − 1.43·59-s − 0.542·61-s + 1.82·65-s − 0.923·67-s + 0.446·71-s + 0.423·73-s − 0.298·77-s − 0.503·79-s − 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.420814716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.420814716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 + 7.56T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−8.587287418001297337183531412402, −7.40130183663487694173232031749, −6.56471656679738190925103441178, −6.22370627862026535303519004543, −5.39045758100929636203926446832, −4.88233114689324812183368757222, −3.44019574146790754690723858989, −2.99936024936045281814387860215, −1.66296625781287452626093035839, −1.19450046447890827961442499760,
1.19450046447890827961442499760, 1.66296625781287452626093035839, 2.99936024936045281814387860215, 3.44019574146790754690723858989, 4.88233114689324812183368757222, 5.39045758100929636203926446832, 6.22370627862026535303519004543, 6.56471656679738190925103441178, 7.40130183663487694173232031749, 8.587287418001297337183531412402
