| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.749 − 0.649i)3-s + (0.841 − 0.540i)4-s + (0.540 + 3.76i)5-s + (−0.536 + 0.834i)6-s + (−2.00 − 1.73i)7-s + (−0.654 + 0.755i)8-s + (−0.286 + 1.99i)9-s + (−1.57 − 3.45i)10-s + (−1.64 + 5.59i)11-s + (0.279 − 0.951i)12-s + (−0.519 + 0.237i)13-s + (2.40 + 1.09i)14-s + (2.84 + 2.46i)15-s + (0.415 − 0.909i)16-s + (−5.39 − 3.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.432 − 0.375i)3-s + (0.420 − 0.270i)4-s + (0.241 + 1.68i)5-s + (−0.218 + 0.340i)6-s + (−0.756 − 0.654i)7-s + (−0.231 + 0.267i)8-s + (−0.0956 + 0.665i)9-s + (−0.499 − 1.09i)10-s + (−0.495 + 1.68i)11-s + (0.0806 − 0.274i)12-s + (−0.144 + 0.0658i)13-s + (0.643 + 0.293i)14-s + (0.735 + 0.637i)15-s + (0.103 − 0.227i)16-s + (−1.30 − 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585503 + 0.687497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585503 + 0.687497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (2.00 + 1.73i)T \) |
| 23 | \( 1 + (-2.13 - 4.29i)T \) |
| good | 3 | \( 1 + (-0.749 + 0.649i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.540 - 3.76i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (1.64 - 5.59i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.519 - 0.237i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (5.39 + 3.46i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.54 + 3.56i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.651 - 0.418i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.47 - 4.74i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.936 + 0.134i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-2.88 + 0.415i)T + (39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.10i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.08iT - 47T^{2} \) |
| 53 | \( 1 + (3.78 + 1.72i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-8.99 + 4.10i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.78 + 5.52i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (1.75 + 5.98i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-9.07 + 2.66i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.13 - 3.31i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (3.57 - 1.63i)T + (51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.38 + 9.63i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (5.73 + 6.62i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.176 - 1.22i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.53773958299607446232413788587, −10.73832874546239252021771729997, −9.971429763196554924787042135665, −9.324982789860708409594506670956, −7.68791233039383719261363306091, −7.08147569253582069332359512801, −6.74074500103343788088194967358, −4.96144587347022024400776347328, −3.05147958513502869004138307055, −2.23328781451991343807718935096,
0.75199228944575992804922708021, 2.72540428876575582311974852728, 3.99834738450402078967751964062, 5.52660194287754256216022405528, 6.32683755031030809803823774410, 8.250008162467269188933970280927, 8.625946485670266031182104596949, 9.313545694195360863099754384768, 10.08900116329798223070686909191, 11.39779560387862245472574167618
