L(s) = 1 | + 2.68·3-s + 2.93·5-s + 1.42·7-s + 4.18·9-s − 0.681·11-s + 3.25·13-s + 7.87·15-s + 4.12·17-s − 7.63·19-s + 3.82·21-s − 4.93·23-s + 3.61·25-s + 3.18·27-s − 7.18·29-s − 4.12·31-s − 1.82·33-s + 4.18·35-s + 9.18·37-s + 8.72·39-s + 8.10·41-s + 8.29·43-s + 12.2·45-s + 9.23·47-s − 4.96·49-s + 11.0·51-s + 4.27·53-s − 2·55-s + ⋯ |
L(s) = 1 | + 1.54·3-s + 1.31·5-s + 0.539·7-s + 1.39·9-s − 0.205·11-s + 0.902·13-s + 2.03·15-s + 1.00·17-s − 1.75·19-s + 0.835·21-s − 1.02·23-s + 0.723·25-s + 0.613·27-s − 1.33·29-s − 0.740·31-s − 0.318·33-s + 0.708·35-s + 1.51·37-s + 1.39·39-s + 1.26·41-s + 1.26·43-s + 1.83·45-s + 1.34·47-s − 0.709·49-s + 1.54·51-s + 0.586·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.070524011\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.070524011\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 83 | \( 1 + T \) |
good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 0.681T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 + 7.18T + 29T^{2} \) |
| 31 | \( 1 + 4.12T + 31T^{2} \) |
| 37 | \( 1 - 9.18T + 37T^{2} \) |
| 41 | \( 1 - 8.10T + 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.098918120259387186658883425752, −7.86775064825473795844224944560, −6.82816294614809845405581083772, −5.88997200090046939207831160811, −5.52428462396018966633493853273, −4.12016787944053846864777342222, −3.80015857322007611315831225415, −2.41023467391733472695549287926, −2.24937978989312901191692980739, −1.24943082899575502134535323756,
1.24943082899575502134535323756, 2.24937978989312901191692980739, 2.41023467391733472695549287926, 3.80015857322007611315831225415, 4.12016787944053846864777342222, 5.52428462396018966633493853273, 5.88997200090046939207831160811, 6.82816294614809845405581083772, 7.86775064825473795844224944560, 8.098918120259387186658883425752