L(s) = 1 | + 3-s − 5-s − 5.16i·7-s + 9-s + 4.48i·11-s + 1.15i·13-s − 15-s + 0.995·17-s + (3.36 + 2.77i)19-s − 5.16i·21-s + 2.60i·23-s + 25-s + 27-s + 9.93i·29-s + 2.14·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.95i·7-s + 0.333·9-s + 1.35i·11-s + 0.319i·13-s − 0.258·15-s + 0.241·17-s + (0.770 + 0.637i)19-s − 1.12i·21-s + 0.542i·23-s + 0.200·25-s + 0.192·27-s + 1.84i·29-s + 0.385·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.140943868\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.140943868\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-3.36 - 2.77i)T \) |
good | 7 | \( 1 + 5.16iT - 7T^{2} \) |
| 11 | \( 1 - 4.48iT - 11T^{2} \) |
| 13 | \( 1 - 1.15iT - 13T^{2} \) |
| 17 | \( 1 - 0.995T + 17T^{2} \) |
| 23 | \( 1 - 2.60iT - 23T^{2} \) |
| 29 | \( 1 - 9.93iT - 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 7.02iT - 37T^{2} \) |
| 41 | \( 1 - 11.4iT - 41T^{2} \) |
| 43 | \( 1 - 3.55iT - 43T^{2} \) |
| 47 | \( 1 + 5.00iT - 47T^{2} \) |
| 53 | \( 1 + 5.43iT - 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 - 8.46T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 - 3.86T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−8.107654560409285007296308120052, −7.57309477673236485114981850807, −7.13442210877204424058154215067, −6.55761002903052828014196384435, −5.13149350976694526274392363926, −4.50788042425906579548644475755, −3.76954874744139196009541554644, −3.24518908493037919369352915500, −1.82412695720394957815289226249, −1.00958915331023038534414514177,
0.65453354341484917337886276538, 2.17688773866185915701699143212, 2.85329469955128846841478590466, 3.45399329544805269992229774555, 4.58183185210884014583464211954, 5.49058539777091259844223134237, 5.95802508672475329778480014282, 6.79642649645390023307830683421, 7.991594662138008749458788251660, 8.187590113262421318020434635528