| L(s) = 1 | + (10.3 + 11.6i)3-s + 68.3i·5-s − 220. i·7-s + (−28.4 + 241. i)9-s − 127.·11-s + 822.·13-s + (−796. + 707. i)15-s + 1.61e3i·17-s − 1.81e3i·19-s + (2.57e3 − 2.28e3i)21-s − 1.27e3·23-s − 1.54e3·25-s + (−3.10e3 + 2.16e3i)27-s + 7.51e3i·29-s + 8.58e3i·31-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.747i)3-s + 1.22i·5-s − 1.70i·7-s + (−0.117 + 0.993i)9-s − 0.316·11-s + 1.34·13-s + (−0.913 + 0.812i)15-s + 1.35i·17-s − 1.15i·19-s + (1.27 − 1.13i)21-s − 0.501·23-s − 0.493·25-s + (−0.820 + 0.572i)27-s + 1.65i·29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.245784377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245784377\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.3 - 11.6i)T \) |
| good | 5 | \( 1 - 68.3iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 220. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 127.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 822.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.61e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.81e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.58e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 4.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.95e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.46e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.40e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.95e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−10.63289925004923386061436956004, −10.37303346836166206063114483622, −9.038873454896278694151586419425, −8.050704376774431547308049740429, −7.15780388878131412063924300183, −6.30204691878273157657354421475, −4.67816063522405711070851810720, −3.67717600460483943686411424378, −3.07299246096864803575267570677, −1.40792197429372111931596660202,
0.50819555382624229345082440689, 1.75834078170040898084427934303, 2.72032850922344536453983343082, 4.14547521896134025658967736970, 5.61627691353200220735277396899, 6.08262656536388520469957768771, 7.72999187134396042362293446179, 8.417607452660022715411283621173, 9.031730929555155601453436927734, 9.756886618034130452698183647006
