| L(s) = 1 | − 2.47·5-s − 2.55·7-s + 0.669·11-s + 4.08·13-s − 6.44·17-s − 6.44·19-s − 2.82·23-s + 1.11·25-s − 4.35·29-s + 6.55·31-s + 6.32·35-s − 3.85·37-s − 0.788·41-s + 0.550·43-s + 2.82·47-s − 0.458·49-s − 3.64·53-s − 1.65·55-s − 5.65·59-s + 6.20·61-s − 10.1·65-s + 2.99·67-s − 5.11·71-s − 14.7·73-s − 1.71·77-s + 6.31·79-s − 0.907·83-s + ⋯ |
| L(s) = 1 | − 1.10·5-s − 0.966·7-s + 0.201·11-s + 1.13·13-s − 1.56·17-s − 1.47·19-s − 0.589·23-s + 0.223·25-s − 0.808·29-s + 1.17·31-s + 1.06·35-s − 0.634·37-s − 0.123·41-s + 0.0840·43-s + 0.412·47-s − 0.0654·49-s − 0.500·53-s − 0.223·55-s − 0.736·59-s + 0.794·61-s − 1.25·65-s + 0.366·67-s − 0.607·71-s − 1.72·73-s − 0.195·77-s + 0.711·79-s − 0.0996·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5641433064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5641433064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.669T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 3.85T + 37T^{2} \) |
| 41 | \( 1 + 0.788T + 41T^{2} \) |
| 43 | \( 1 - 0.550T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 6.20T + 61T^{2} \) |
| 67 | \( 1 - 2.99T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 6.31T + 79T^{2} \) |
| 83 | \( 1 + 0.907T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.77669491031826133470448943520, −6.90147425322370818587199689724, −6.41671959151686278339198735493, −5.93294349719607321164754551199, −4.69994492696638575884327230411, −4.06253047098692020774931688828, −3.66393692288977074652037825721, −2.72491448978443404533273645287, −1.75534487576057498731978088792, −0.34810070370882689301693943648,
0.34810070370882689301693943648, 1.75534487576057498731978088792, 2.72491448978443404533273645287, 3.66393692288977074652037825721, 4.06253047098692020774931688828, 4.69994492696638575884327230411, 5.93294349719607321164754551199, 6.41671959151686278339198735493, 6.90147425322370818587199689724, 7.77669491031826133470448943520
