| L(s) = 1 | − 0.385·3-s − 5-s − 2.85·9-s + 5.19·11-s + 6.04·13-s + 0.385·15-s + 3.19·17-s + 7.52·19-s − 1.70·23-s + 25-s + 2.25·27-s + 7.04·29-s − 6.51·31-s − 2.00·33-s − 2.85·37-s − 2.32·39-s + 41-s + 6.58·43-s + 2.85·45-s + 5.19·47-s − 1.23·51-s − 4.04·53-s − 5.19·55-s − 2.89·57-s − 7.28·59-s − 6.65·61-s − 6.04·65-s + ⋯ |
| L(s) = 1 | − 0.222·3-s − 0.447·5-s − 0.950·9-s + 1.56·11-s + 1.67·13-s + 0.0994·15-s + 0.774·17-s + 1.72·19-s − 0.355·23-s + 0.200·25-s + 0.433·27-s + 1.30·29-s − 1.16·31-s − 0.348·33-s − 0.468·37-s − 0.373·39-s + 0.156·41-s + 1.00·43-s + 0.425·45-s + 0.757·47-s − 0.172·51-s − 0.555·53-s − 0.699·55-s − 0.383·57-s − 0.947·59-s − 0.852·61-s − 0.749·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186671794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186671794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.385T + 3T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 - 7.04T + 29T^{2} \) |
| 31 | \( 1 + 6.51T + 31T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 + 6.65T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 + 4.64T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 0.389T + 97T^{2} \) |
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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
−7.896724444710825035609445603782, −7.14189735931559917085671179920, −6.34856202138187031916834190790, −5.85744198498990919507809101559, −5.18382869625383084525518331726, −4.10428262769573852703206418253, −3.55139697775007564592592829462, −2.94783942382580821961463878813, −1.47102915297273095751824368482, −0.837721657859855491855529389603,
0.837721657859855491855529389603, 1.47102915297273095751824368482, 2.94783942382580821961463878813, 3.55139697775007564592592829462, 4.10428262769573852703206418253, 5.18382869625383084525518331726, 5.85744198498990919507809101559, 6.34856202138187031916834190790, 7.14189735931559917085671179920, 7.896724444710825035609445603782
