| L(s) = 1 | + 1.61·5-s − 5.19·7-s + 2.56·11-s − 0.161·13-s + 6.00·17-s + 4.13·19-s − 8.13·23-s − 2.38·25-s + 1.20·29-s + 6.04·31-s − 8.40·35-s + 1.62·37-s + 4.16·41-s + 3.31·43-s − 9.89·47-s + 19.9·49-s + 8.24·53-s + 4.15·55-s + 12.5·59-s + 8.52·61-s − 0.261·65-s + 12.2·67-s − 7.09·71-s − 0.613·73-s − 13.3·77-s + 0.259·79-s + 8.20·83-s + ⋯ |
| L(s) = 1 | + 0.723·5-s − 1.96·7-s + 0.773·11-s − 0.0448·13-s + 1.45·17-s + 0.947·19-s − 1.69·23-s − 0.476·25-s + 0.224·29-s + 1.08·31-s − 1.41·35-s + 0.267·37-s + 0.649·41-s + 0.505·43-s − 1.44·47-s + 2.85·49-s + 1.13·53-s + 0.559·55-s + 1.63·59-s + 1.09·61-s − 0.0324·65-s + 1.50·67-s − 0.841·71-s − 0.0718·73-s − 1.51·77-s + 0.0292·79-s + 0.900·83-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
| \(L(1)\) |
\(\approx\) |
\(1.662497080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662497080\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 5.19T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 0.161T + 13T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 + 8.13T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 8.52T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 + 0.613T + 73T^{2} \) |
| 79 | \( 1 - 0.259T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 + 5.96T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.603646120000484050928274791472, −8.503729988641908026837129499354, −7.52875906653053290179275474468, −6.63966948750824709811595031032, −6.04431489050574554022551705828, −5.48628362286538099301910801409, −3.98674026932909129837867411400, −3.34898594961331378589634839737, −2.35110153815255004242710735310, −0.870602071290114221995425215203,
0.870602071290114221995425215203, 2.35110153815255004242710735310, 3.34898594961331378589634839737, 3.98674026932909129837867411400, 5.48628362286538099301910801409, 6.04431489050574554022551705828, 6.63966948750824709811595031032, 7.52875906653053290179275474468, 8.503729988641908026837129499354, 9.603646120000484050928274791472
