| L(s) = 1 | + 2.16·7-s − 3·9-s + 0.162·11-s − 13-s + 7.32·17-s + 2·19-s + 4.32·23-s + 29-s + 0.162·31-s + 6·37-s − 12.3·43-s − 8.16·47-s − 2.32·49-s − 5.32·53-s + 5.83·59-s + 3.32·61-s − 6.48·63-s + 2.16·67-s − 10.6·71-s + 10.6·73-s + 0.350·77-s + 8.32·79-s + 9·81-s − 9.83·83-s + 16.6·89-s − 2.16·91-s + 10·97-s + ⋯ |
| L(s) = 1 | + 0.817·7-s − 9-s + 0.0489·11-s − 0.277·13-s + 1.77·17-s + 0.458·19-s + 0.901·23-s + 0.185·29-s + 0.0291·31-s + 0.986·37-s − 1.87·43-s − 1.19·47-s − 0.332·49-s − 0.731·53-s + 0.760·59-s + 0.425·61-s − 0.817·63-s + 0.264·67-s − 1.26·71-s + 1.24·73-s + 0.0399·77-s + 0.936·79-s + 81-s − 1.07·83-s + 1.76·89-s − 0.226·91-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.110981392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.110981392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 - 0.162T + 11T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 0.162T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 8.16T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 - 3.32T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.056717813756388712582981522410, −7.75187948157186546264118190184, −6.77818412018608283728457840652, −5.96094048166934299531973256435, −5.17964157763558050679442940894, −4.81481378281062737945821565481, −3.49171559641011948347901517548, −2.99266923060370776969883781549, −1.83920691448561932679691443032, −0.810398906887766923592802347512,
0.810398906887766923592802347512, 1.83920691448561932679691443032, 2.99266923060370776969883781549, 3.49171559641011948347901517548, 4.81481378281062737945821565481, 5.17964157763558050679442940894, 5.96094048166934299531973256435, 6.77818412018608283728457840652, 7.75187948157186546264118190184, 8.056717813756388712582981522410
