| L(s) = 1 | − 4.37·5-s + 7-s + 2.64·11-s + 4·13-s + 3.46·17-s + 3.58·19-s + 3.46·23-s + 14.1·25-s + 1.82·29-s − 9.16·31-s − 4.37·35-s + 3·37-s + 4.37·41-s + 8.58·43-s − 2.74·47-s + 49-s − 8.66·53-s − 11.5·55-s − 3.46·59-s + 2.41·61-s − 17.5·65-s + 0.582·67-s − 11.4·71-s − 3.16·73-s + 2.64·77-s + 8.58·79-s − 6.20·83-s + ⋯ |
| L(s) = 1 | − 1.95·5-s + 0.377·7-s + 0.797·11-s + 1.10·13-s + 0.840·17-s + 0.821·19-s + 0.722·23-s + 2.83·25-s + 0.339·29-s − 1.64·31-s − 0.739·35-s + 0.493·37-s + 0.683·41-s + 1.30·43-s − 0.399·47-s + 0.142·49-s − 1.18·53-s − 1.56·55-s − 0.450·59-s + 0.309·61-s − 2.17·65-s + 0.0711·67-s − 1.35·71-s − 0.370·73-s + 0.301·77-s + 0.965·79-s − 0.681·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743683202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743683202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.82T + 29T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 - 0.582T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 - 8.75T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−7.59507708554501066928181989801, −7.36683482026774018574235930899, −6.47499132561472873517379985815, −5.64462297184992318000121241397, −4.78232419579792413336737874362, −4.10275699757139131967432156048, −3.54255488828347524246264408495, −2.99263690795249172503381424695, −1.42086187153501670610053224735, −0.71541452664936098332074175503,
0.71541452664936098332074175503, 1.42086187153501670610053224735, 2.99263690795249172503381424695, 3.54255488828347524246264408495, 4.10275699757139131967432156048, 4.78232419579792413336737874362, 5.64462297184992318000121241397, 6.47499132561472873517379985815, 7.36683482026774018574235930899, 7.59507708554501066928181989801
