| L(s) = 1 | + (0.103 − 1.99i)2-s + (0.204 − 2.99i)3-s + (−3.97 − 0.414i)4-s + (1.50 − 8.54i)5-s + (−5.95 − 0.719i)6-s + (7.53 + 8.98i)7-s + (−1.24 + 7.90i)8-s + (−8.91 − 1.22i)9-s + (−16.9 − 3.89i)10-s + (−2.27 + 0.400i)11-s + (−2.05 + 11.8i)12-s + (−4.42 + 1.61i)13-s + (18.7 − 14.1i)14-s + (−25.2 − 6.25i)15-s + (15.6 + 3.30i)16-s + (6.77 − 11.7i)17-s + ⋯ |
| L(s) = 1 | + (0.0519 − 0.998i)2-s + (0.0681 − 0.997i)3-s + (−0.994 − 0.103i)4-s + (0.301 − 1.70i)5-s + (−0.992 − 0.119i)6-s + (1.07 + 1.28i)7-s + (−0.155 + 0.987i)8-s + (−0.990 − 0.136i)9-s + (−1.69 − 0.389i)10-s + (−0.206 + 0.0364i)11-s + (−0.171 + 0.985i)12-s + (−0.340 + 0.123i)13-s + (1.33 − 1.00i)14-s + (−1.68 − 0.417i)15-s + (0.978 + 0.206i)16-s + (0.398 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.185403 - 1.34888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185403 - 1.34888i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.103 + 1.99i)T \) |
| 3 | \( 1 + (-0.204 + 2.99i)T \) |
| good | 5 | \( 1 + (-1.50 + 8.54i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-7.53 - 8.98i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.27 - 0.400i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.42 - 1.61i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-6.77 + 11.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.9 + 8.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (3.20 - 3.82i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-34.0 - 12.4i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (0.283 - 0.337i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-18.9 + 32.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.35 + 0.858i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (17.5 - 3.10i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-5.64 - 6.72i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 41.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-25.5 - 4.51i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-51.1 + 42.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-32.3 - 88.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (100. + 57.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-22.5 - 39.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (19.8 - 54.4i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (22.8 - 62.8i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-54.3 - 94.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.2 - 171. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.67735916252695546940214831989, −12.09008931735244068457333650638, −11.43879902298206649441700257098, −9.491246240086271544982950788184, −8.721270409268901915090594678172, −7.958050098243106537253281778602, −5.52619321784464094312872655634, −4.92463601032980749511327185508, −2.40330119433818116995004636559, −1.13472372025957606102870251351,
3.38098767903631055054503859858, 4.63185123995302415523007571009, 6.05816030898693731973279795830, 7.32888867956536924109255529241, 8.191763065096214810380661403607, 10.03687474956835239971929586347, 10.34554723106469807333592180503, 11.51109344755795346987874474487, 13.61873247853369287727260037366, 14.35651130049019766774453793330
