| L(s) = 1 | − i·7-s + 3·9-s + 11-s − 2i·13-s + 4i·17-s + 2·19-s − 5i·23-s − 29-s − 2·31-s + 3i·37-s + 12·41-s − 11i·43-s + 2i·47-s − 49-s − 6i·53-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s + 9-s + 0.301·11-s − 0.554i·13-s + 0.970i·17-s + 0.458·19-s − 1.04i·23-s − 0.185·29-s − 0.359·31-s + 0.493i·37-s + 1.87·41-s − 1.67i·43-s + 0.291i·47-s − 0.142·49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.848585316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848585316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 11iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.599061180425758809174744402252, −8.678923886928553490258809363353, −7.84225347065163496641354453421, −7.09184273427207850064323285530, −6.30961418212361023393804338565, −5.31246900451201230939971897737, −4.27705929900894322708507356213, −3.59575598684042770727745005932, −2.19880435768382827313471797407, −0.929880260616596768860034320528,
1.19435034322577964247896542003, 2.40480825458192401725560936454, 3.63389025602634098694874082133, 4.52952872578545709568449537331, 5.43204864951465508273921149027, 6.39925925855325424010351243521, 7.28583188790213835075936085875, 7.81223633686724022463121525964, 9.206859942768083291343977409771, 9.358228634353551045451433032501
