| L(s) = 1 | + 2.01·5-s + 1.72·7-s + 2.36·11-s + 2.55·13-s − 1.95·17-s + 4.57·19-s − 0.627·23-s − 0.946·25-s + 6.00·29-s − 7.74·31-s + 3.47·35-s + 9.29·37-s + 1.44·41-s + 4.99·43-s − 10.5·47-s − 4.02·49-s − 8.02·53-s + 4.76·55-s − 10.1·59-s + 14.1·61-s + 5.13·65-s − 4.22·67-s + 3.62·71-s + 13.1·73-s + 4.07·77-s + 5.84·79-s − 13.7·83-s + ⋯ |
| L(s) = 1 | + 0.900·5-s + 0.651·7-s + 0.713·11-s + 0.707·13-s − 0.473·17-s + 1.05·19-s − 0.130·23-s − 0.189·25-s + 1.11·29-s − 1.39·31-s + 0.586·35-s + 1.52·37-s + 0.226·41-s + 0.762·43-s − 1.53·47-s − 0.575·49-s − 1.10·53-s + 0.642·55-s − 1.32·59-s + 1.81·61-s + 0.637·65-s − 0.515·67-s + 0.430·71-s + 1.54·73-s + 0.464·77-s + 0.657·79-s − 1.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.474315959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474315959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 0.627T + 23T^{2} \) |
| 29 | \( 1 - 6.00T + 29T^{2} \) |
| 31 | \( 1 + 7.74T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 4.22T + 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
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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
−9.422272845972986307595550388008, −8.410101931335739969307132906310, −7.72302481630283822229717347949, −6.66444552658886800654732702227, −6.04567375176591561235287479396, −5.21621219697597208044028828708, −4.32772970681858336980557645532, −3.28395652573887503234904504243, −2.06701866747887795024498104549, −1.17812892079680341174107210890,
1.17812892079680341174107210890, 2.06701866747887795024498104549, 3.28395652573887503234904504243, 4.32772970681858336980557645532, 5.21621219697597208044028828708, 6.04567375176591561235287479396, 6.66444552658886800654732702227, 7.72302481630283822229717347949, 8.410101931335739969307132906310, 9.422272845972986307595550388008
