| L(s) = 1 | − 2i·5-s + 7-s − 2i·11-s − 6i·13-s + 6·17-s − 8i·19-s − 8·23-s + 25-s + 8·31-s − 2i·35-s + 4i·37-s − 6·41-s + 6i·43-s − 12·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + 0.377·7-s − 0.603i·11-s − 1.66i·13-s + 1.45·17-s − 1.83i·19-s − 1.66·23-s + 0.200·25-s + 1.43·31-s − 0.338i·35-s + 0.657i·37-s − 0.937·41-s + 0.914i·43-s − 1.75·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689982277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689982277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.107806387445461734908194411798, −7.81432805183976559464225862439, −6.60557678119049794964677576086, −5.83059751364762622630481314685, −5.09679738832542045696119401124, −4.62278395114556380682361410572, −3.38820289370264574518715979778, −2.74756526346496577955245762443, −1.30419564962140819913175798835, −0.50700070140376150295088311047,
1.54555916467249569238410133011, 2.18741415405144544660374849034, 3.44120147904950181444688071969, 4.01516792086914401998502170719, 4.94652862760079836649269118425, 5.91747212544699068338250343257, 6.52351628240022775683468020893, 7.24645089593918393175211757276, 7.969095571795307483864421522974, 8.523314379397523693242547652349
