| L(s) = 1 | − 1.89·5-s − 1.71·7-s − 0.839·11-s − 1.94·13-s − 4.19·17-s − 1.07·19-s + 0.340·23-s − 1.42·25-s − 4.65·29-s − 9.22·31-s + 3.23·35-s − 3.13·37-s − 5.59·41-s − 1.01·43-s + 10.4·47-s − 4.07·49-s + 9.10·53-s + 1.58·55-s − 59-s − 0.592·61-s + 3.68·65-s + 1.37·67-s + 11.2·71-s + 8.50·73-s + 1.43·77-s + 11.4·79-s + 3.28·83-s + ⋯ |
| L(s) = 1 | − 0.845·5-s − 0.646·7-s − 0.253·11-s − 0.540·13-s − 1.01·17-s − 0.245·19-s + 0.0710·23-s − 0.284·25-s − 0.865·29-s − 1.65·31-s + 0.547·35-s − 0.515·37-s − 0.874·41-s − 0.154·43-s + 1.53·47-s − 0.581·49-s + 1.25·53-s + 0.214·55-s − 0.130·59-s − 0.0758·61-s + 0.457·65-s + 0.167·67-s + 1.33·71-s + 0.995·73-s + 0.163·77-s + 1.28·79-s + 0.360·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5281255120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5281255120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 0.839T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 - 0.340T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 61 | \( 1 + 0.592T + 61T^{2} \) |
| 67 | \( 1 - 1.37T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.76125566303256929135732316050, −7.04508938534087949274509090551, −6.64582317320061740047852104389, −5.61466067167430585350670786725, −5.06045512668071101849036520766, −3.99801947507172500894477918971, −3.71336521397379873860735756394, −2.65945058312234188839960391726, −1.87103303724676207073883998412, −0.33540536894538864999002420405,
0.33540536894538864999002420405, 1.87103303724676207073883998412, 2.65945058312234188839960391726, 3.71336521397379873860735756394, 3.99801947507172500894477918971, 5.06045512668071101849036520766, 5.61466067167430585350670786725, 6.64582317320061740047852104389, 7.04508938534087949274509090551, 7.76125566303256929135732316050
