L(s) = 1 | + 7.95i·3-s − 46.0·5-s + 57.5i·7-s + 17.7·9-s − 181. i·11-s − 1.33·13-s − 366. i·15-s − 62.6·17-s − 340. i·19-s − 457.·21-s − 447. i·23-s + 1.49e3·25-s + 785. i·27-s − 497.·29-s − 444. i·31-s + ⋯ |
L(s) = 1 | + 0.883i·3-s − 1.84·5-s + 1.17i·7-s + 0.219·9-s − 1.50i·11-s − 0.00787·13-s − 1.62i·15-s − 0.216·17-s − 0.942i·19-s − 1.03·21-s − 0.845i·23-s + 2.39·25-s + 1.07i·27-s − 0.591·29-s − 0.462i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.279104 - 0.279104i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.279104 - 0.279104i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 7.95iT - 81T^{2} \) |
| 5 | \( 1 + 46.0T + 625T^{2} \) |
| 7 | \( 1 - 57.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 181. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 1.33T + 2.85e4T^{2} \) |
| 17 | \( 1 + 62.6T + 8.35e4T^{2} \) |
| 19 | \( 1 + 340. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 447. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 497.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 444. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 687.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 3.07e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 307. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.04e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.97e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.87e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.53e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.96e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 910. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.40e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.52e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 9.68e3T + 8.85e7T^{2} \) |
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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
−12.03789528387323464344220348195, −11.42283952941994371142041671565, −10.51808375644252358076058568797, −8.885832927718517112427222151635, −8.439044410176831508738591985833, −6.98859463466360932806466084785, −5.33135325551397390549649205437, −4.12668559076560754871851564261, −3.09235504121407914201739380025, −0.17803050075216807362781695384,
1.40816662538624578850880375569, 3.68823581047874846273907781867, 4.58576668412179589228782500321, 6.92334815481932108174987205538, 7.38957944258554585555548330264, 8.132525247855342748138763867003, 9.894576490420777640112381019454, 11.01856183372077025488574068452, 12.12391406794645191891650151817, 12.61231694005436845457669512069