| L(s) = 1 | − 5-s − 1.56·7-s + 3.12·11-s + 2·13-s − 3.56·17-s − 2·19-s − 23-s + 25-s − 6.68·29-s + 4.68·31-s + 1.56·35-s + 2.43·37-s + 2.68·41-s − 4·47-s − 4.56·49-s − 7.56·53-s − 3.12·55-s − 3.56·59-s + 9.12·61-s − 2·65-s + 8.68·67-s − 0.438·71-s − 4.24·73-s − 4.87·77-s − 2.87·79-s − 12.6·83-s + 3.56·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.590·7-s + 0.941·11-s + 0.554·13-s − 0.863·17-s − 0.458·19-s − 0.208·23-s + 0.200·25-s − 1.24·29-s + 0.841·31-s + 0.263·35-s + 0.400·37-s + 0.419·41-s − 0.583·47-s − 0.651·49-s − 1.03·53-s − 0.421·55-s − 0.463·59-s + 1.16·61-s − 0.248·65-s + 1.06·67-s − 0.0520·71-s − 0.496·73-s − 0.555·77-s − 0.323·79-s − 1.39·83-s + 0.386·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 3.56T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 + 0.438T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.125111924440308208641931118910, −7.26794457891808745717882071468, −6.47250147297528529127444795157, −6.12097262139449166300047833257, −4.95639415509300466858036466286, −4.09108229285627994637017246811, −3.56941221185371986249454273541, −2.50725038985008108395192898733, −1.36475244999116297851070167764, 0,
1.36475244999116297851070167764, 2.50725038985008108395192898733, 3.56941221185371986249454273541, 4.09108229285627994637017246811, 4.95639415509300466858036466286, 6.12097262139449166300047833257, 6.47250147297528529127444795157, 7.26794457891808745717882071468, 8.125111924440308208641931118910
