| L(s) = 1 | + 2.18·5-s + 4.85·7-s + 11-s + 13-s − 3.92·17-s − 2.48·19-s + 6.10·23-s − 0.236·25-s + 6.66·29-s − 8.02·31-s + 10.5·35-s − 1.25·37-s + 2.85·41-s − 1.92·43-s + 8.95·47-s + 16.5·49-s + 7.33·53-s + 2.18·55-s + 1.13·59-s + 4.95·61-s + 2.18·65-s + 3.04·67-s − 10.2·71-s + 2.12·73-s + 4.85·77-s + 10.2·79-s − 6.20·83-s + ⋯ |
| L(s) = 1 | + 0.976·5-s + 1.83·7-s + 0.301·11-s + 0.277·13-s − 0.950·17-s − 0.570·19-s + 1.27·23-s − 0.0472·25-s + 1.23·29-s − 1.44·31-s + 1.79·35-s − 0.205·37-s + 0.445·41-s − 0.292·43-s + 1.30·47-s + 2.36·49-s + 1.00·53-s + 0.294·55-s + 0.147·59-s + 0.634·61-s + 0.270·65-s + 0.372·67-s − 1.22·71-s + 0.248·73-s + 0.552·77-s + 1.15·79-s − 0.681·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.229836775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.229836775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 + 4.30T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−8.435670281935510611843415948311, −7.44938950717695252164650908936, −6.83626953472287045370225459177, −5.94295837639668339578029734841, −5.27866998927765262082192536507, −4.64673023876810134162869466458, −3.90366345384859803153865845007, −2.52226535648290312354664401310, −1.90252735602972131921879023585, −1.06169673756558190021066408243,
1.06169673756558190021066408243, 1.90252735602972131921879023585, 2.52226535648290312354664401310, 3.90366345384859803153865845007, 4.64673023876810134162869466458, 5.27866998927765262082192536507, 5.94295837639668339578029734841, 6.83626953472287045370225459177, 7.44938950717695252164650908936, 8.435670281935510611843415948311
