| L(s) = 1 | + 6.25i·2-s − 21.3i·3-s − 7.11·4-s + 133.·6-s − 116. i·7-s + 155. i·8-s − 213.·9-s − 259.·11-s + 152. i·12-s + 169i·13-s + 727.·14-s − 1.20e3·16-s − 876. i·17-s − 1.33e3i·18-s + 921.·19-s + ⋯ |
| L(s) = 1 | + 1.10i·2-s − 1.37i·3-s − 0.222·4-s + 1.51·6-s − 0.897i·7-s + 0.859i·8-s − 0.880·9-s − 0.645·11-s + 0.305i·12-s + 0.277i·13-s + 0.992·14-s − 1.17·16-s − 0.735i·17-s − 0.973i·18-s + 0.585·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5452958553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5452958553\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 - 6.25iT - 32T^{2} \) |
| 3 | \( 1 + 21.3iT - 243T^{2} \) |
| 7 | \( 1 + 116. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 259.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 876. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 921.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.05e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 781.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.81e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.18e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.66e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 763. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.73e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.51e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.97e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.16e4iT - 8.58e9T^{2} \) |
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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
−10.46965591068427910740718575786, −9.045738676594913244141566878022, −7.87439955395656975289081220226, −7.42921783309942903585018455991, −6.75149923094796152809336867777, −5.83703545538146828027550499330, −4.65009716430432302613886380304, −2.78728132722754434436462039183, −1.49729564351410859940226877168, −0.13160402507789351749944687784,
1.74781311710972457969136815359, 2.99868734844195749244487282668, 3.76098703836354212517163431541, 4.98317167694896253399841079879, 5.91314089536925104845374446175, 7.52621151723656881452324087281, 8.919081058627703981566489061273, 9.536024327785707647445825287030, 10.43666230240860933508718463175, 10.92260517958753307560318100737
