| L(s) = 1 | + 5-s − 0.732·7-s − 1.73·11-s − 1.46·13-s + 1.26·17-s − 2.46·19-s − 3.46·23-s + 25-s + 4.26·29-s + 7.92·31-s − 0.732·35-s + 4.19·37-s + 0.803·41-s − 6.73·43-s − 4.73·47-s − 6.46·49-s + 10.7·53-s − 1.73·55-s − 4.26·59-s − 4·61-s − 1.46·65-s + 14.3·67-s − 0.803·71-s + 10.1·73-s + 1.26·77-s − 6.39·79-s + 9.12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.276·7-s − 0.522·11-s − 0.406·13-s + 0.307·17-s − 0.565·19-s − 0.722·23-s + 0.200·25-s + 0.792·29-s + 1.42·31-s − 0.123·35-s + 0.689·37-s + 0.125·41-s − 1.02·43-s − 0.690·47-s − 0.923·49-s + 1.47·53-s − 0.233·55-s − 0.555·59-s − 0.512·61-s − 0.181·65-s + 1.75·67-s − 0.0953·71-s + 1.19·73-s + 0.144·77-s − 0.719·79-s + 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798507272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798507272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 0.803T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 0.803T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
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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
−8.146918167600598776108243481729, −7.29933172394859758977834813784, −6.46533270882855706074519438040, −6.05230079863138445783003937212, −5.08307898543234930119941655178, −4.55607684441143564552200608143, −3.52165234018459590788960762004, −2.69163737490376644624863095695, −1.95117033731855829177907842489, −0.67944127409426459877210398832,
0.67944127409426459877210398832, 1.95117033731855829177907842489, 2.69163737490376644624863095695, 3.52165234018459590788960762004, 4.55607684441143564552200608143, 5.08307898543234930119941655178, 6.05230079863138445783003937212, 6.46533270882855706074519438040, 7.29933172394859758977834813784, 8.146918167600598776108243481729
