| L(s) = 1 | + (2.37 − 3.26i)2-s + (4.81 − 14.8i)3-s + (14.7 + 45.3i)4-s + (26.5 − 19.2i)5-s + (−36.9 − 50.9i)6-s + (457. − 148. i)7-s + (428. + 139. i)8-s + (−196. − 142. i)9-s − 132. i·10-s + (−1.20e3 − 571. i)11-s + 743.·12-s + (2.08e3 − 2.86e3i)13-s + (599. − 1.84e3i)14-s + (−158. − 486. i)15-s + (−997. + 724. i)16-s + (1.76e3 + 2.42e3i)17-s + ⋯ |
| L(s) = 1 | + (0.296 − 0.408i)2-s + (0.178 − 0.549i)3-s + (0.230 + 0.708i)4-s + (0.212 − 0.154i)5-s + (−0.171 − 0.235i)6-s + (1.33 − 0.433i)7-s + (0.837 + 0.272i)8-s + (−0.269 − 0.195i)9-s − 0.132i·10-s + (−0.903 − 0.429i)11-s + 0.430·12-s + (0.947 − 1.30i)13-s + (0.218 − 0.672i)14-s + (−0.0468 − 0.144i)15-s + (−0.243 + 0.176i)16-s + (0.359 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.24369 - 0.985056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24369 - 0.985056i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.81 + 14.8i)T \) |
| 11 | \( 1 + (1.20e3 + 571. i)T \) |
| good | 2 | \( 1 + (-2.37 + 3.26i)T + (-19.7 - 60.8i)T^{2} \) |
| 5 | \( 1 + (-26.5 + 19.2i)T + (4.82e3 - 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-457. + 148. i)T + (9.51e4 - 6.91e4i)T^{2} \) |
| 13 | \( 1 + (-2.08e3 + 2.86e3i)T + (-1.49e6 - 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-1.76e3 - 2.42e3i)T + (-7.45e6 + 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-4.84e3 - 1.57e3i)T + (3.80e7 + 2.76e7i)T^{2} \) |
| 23 | \( 1 + 1.31e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.91e4 - 9.48e3i)T + (4.81e8 - 3.49e8i)T^{2} \) |
| 31 | \( 1 + (1.16e4 + 8.48e3i)T + (2.74e8 + 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-2.00e4 - 6.17e4i)T + (-2.07e9 + 1.50e9i)T^{2} \) |
| 41 | \( 1 + (7.87e4 + 2.55e4i)T + (3.84e9 + 2.79e9i)T^{2} \) |
| 43 | \( 1 - 1.06e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-5.93e4 + 1.82e5i)T + (-8.72e9 - 6.33e9i)T^{2} \) |
| 53 | \( 1 + (7.32e4 + 5.32e4i)T + (6.84e9 + 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-3.80e4 - 1.17e5i)T + (-3.41e10 + 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.80e5 + 2.48e5i)T + (-1.59e10 + 4.89e10i)T^{2} \) |
| 67 | \( 1 + 1.74e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (4.50e5 - 3.26e5i)T + (3.95e10 - 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-1.14e5 + 3.71e4i)T + (1.22e11 - 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-1.88e4 + 2.59e4i)T + (-7.51e10 - 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-3.57e5 - 4.92e5i)T + (-1.01e11 + 3.10e11i)T^{2} \) |
| 89 | \( 1 - 4.46e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-3.63e5 - 2.63e5i)T + (2.57e11 + 7.92e11i)T^{2} \) |
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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
−15.11685553676605907729710481514, −13.62265051139057014475216248462, −13.00721265454795322551748991807, −11.57327935134968595884485091257, −10.64423371877911259929119246081, −8.273506798865592794426294082282, −7.65155655433842967928349902905, −5.38607559146043154922975467529, −3.37807204370663717514365236533, −1.51570312023299478304334763218,
1.90683899918099609105052366703, 4.58156885996820606546297448794, 5.74018440306477154067611012236, 7.55870948901000494567329719043, 9.204795610098881638090244378997, 10.61618874935102185528550123599, 11.60165473857591868900875432970, 13.72284408700485030392390585347, 14.45613784528188685840054834985, 15.47741721011669638506244636966
