| L(s) = 1 | + 1.64·5-s − 1.01·7-s + 5.21·11-s + 2.55·13-s − 3.56·17-s + 3.09·19-s + 23-s − 2.29·25-s − 1.44·29-s + 1.44·31-s − 1.66·35-s + 6.10·37-s − 4.73·41-s + 11.3·43-s − 0.0221·47-s − 5.97·49-s − 7.66·53-s + 8.57·55-s + 3.28·59-s + 7.21·61-s + 4.20·65-s + 11.4·67-s − 0.736·71-s + 11.6·73-s − 5.26·77-s − 2.12·79-s + 9.21·83-s + ⋯ |
| L(s) = 1 | + 0.736·5-s − 0.382·7-s + 1.57·11-s + 0.708·13-s − 0.864·17-s + 0.709·19-s + 0.208·23-s − 0.458·25-s − 0.268·29-s + 0.259·31-s − 0.281·35-s + 1.00·37-s − 0.739·41-s + 1.72·43-s − 0.00323·47-s − 0.853·49-s − 1.05·53-s + 1.15·55-s + 0.427·59-s + 0.923·61-s + 0.521·65-s + 1.40·67-s − 0.0874·71-s + 1.36·73-s − 0.600·77-s − 0.238·79-s + 1.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132251218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132251218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.0221T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 - 3.28T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 0.736T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.12T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−9.418134157934308153180063701318, −8.800463077767531576557343659063, −7.80043737064005751930431914699, −6.65259535248385408401204668964, −6.33351486464901867062461929631, −5.40267508800869926011251159297, −4.24884199433736633489853412250, −3.47014598135513149805331511785, −2.20683097657931626766559044537, −1.09761180930441354968088151586,
1.09761180930441354968088151586, 2.20683097657931626766559044537, 3.47014598135513149805331511785, 4.24884199433736633489853412250, 5.40267508800869926011251159297, 6.33351486464901867062461929631, 6.65259535248385408401204668964, 7.80043737064005751930431914699, 8.800463077767531576557343659063, 9.418134157934308153180063701318
