| L(s) = 1 | + 5-s + 2.71·7-s + 2.76·11-s − 1.01·13-s − 4.50·17-s + 4.76·19-s + 23-s + 25-s + 8.08·29-s − 5.49·31-s + 2.71·35-s − 10.9·37-s − 2.43·41-s + 5.77·43-s + 11.0·47-s + 0.356·49-s + 8.15·53-s + 2.76·55-s − 1.70·59-s + 3.12·61-s − 1.01·65-s + 5.56·67-s − 2.07·71-s + 9.63·73-s + 7.49·77-s − 3.63·79-s + 11.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.02·7-s + 0.833·11-s − 0.280·13-s − 1.09·17-s + 1.09·19-s + 0.208·23-s + 0.200·25-s + 1.50·29-s − 0.986·31-s + 0.458·35-s − 1.80·37-s − 0.380·41-s + 0.880·43-s + 1.61·47-s + 0.0509·49-s + 1.11·53-s + 0.372·55-s − 0.221·59-s + 0.400·61-s − 0.125·65-s + 0.679·67-s − 0.246·71-s + 1.12·73-s + 0.854·77-s − 0.409·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.844169869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.844169869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 8.15T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 3.63T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 4.13T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.77613123578000845192872578522, −7.03464312280223326771081121724, −6.58358677270305216707555746942, −5.57381143784980566143549943265, −5.08055642214788381092113862290, −4.34634240116177246886416110852, −3.56120644817169924573871883052, −2.50225248407168403152636984477, −1.76520432537489537850685469556, −0.871246311518717853347318950590,
0.871246311518717853347318950590, 1.76520432537489537850685469556, 2.50225248407168403152636984477, 3.56120644817169924573871883052, 4.34634240116177246886416110852, 5.08055642214788381092113862290, 5.57381143784980566143549943265, 6.58358677270305216707555746942, 7.03464312280223326771081121724, 7.77613123578000845192872578522
