| L(s) = 1 | + (−1.37 + 1.45i)2-s + (−2.32 + 1.89i)3-s + (−0.206 − 3.99i)4-s + (2.67 + 7.36i)5-s + (0.442 − 5.98i)6-s + (10.3 − 1.82i)7-s + (6.07 + 5.20i)8-s + (1.78 − 8.82i)9-s + (−14.3 − 6.25i)10-s + (16.0 + 5.85i)11-s + (8.06 + 8.88i)12-s + (5.02 − 5.99i)13-s + (−11.6 + 17.5i)14-s + (−20.2 − 12.0i)15-s + (−15.9 + 1.64i)16-s + (0.0223 − 0.0386i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 + 0.725i)2-s + (−0.773 + 0.633i)3-s + (−0.0515 − 0.998i)4-s + (0.535 + 1.47i)5-s + (0.0738 − 0.997i)6-s + (1.47 − 0.260i)7-s + (0.759 + 0.650i)8-s + (0.198 − 0.980i)9-s + (−1.43 − 0.625i)10-s + (1.46 + 0.532i)11-s + (0.672 + 0.740i)12-s + (0.386 − 0.460i)13-s + (−0.828 + 1.25i)14-s + (−1.34 − 0.800i)15-s + (−0.994 + 0.102i)16-s + (0.00131 − 0.00227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.573227 + 1.00563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.573227 + 1.00563i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 1.45i)T \) |
| 3 | \( 1 + (2.32 - 1.89i)T \) |
| good | 5 | \( 1 + (-2.67 - 7.36i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-10.3 + 1.82i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-16.0 - 5.85i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-5.02 + 5.99i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-0.0223 + 0.0386i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.17 - 14.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (19.2 + 3.39i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (8.26 + 9.84i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-37.6 - 6.64i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (59.4 + 34.2i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.97 - 4.17i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (26.0 + 9.46i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (29.0 - 5.12i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 11.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-60.3 + 21.9i)T + (2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (7.03 - 1.24i)T + (3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-83.6 - 70.2i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-36.8 - 21.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (16.0 + 27.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (100. + 119. i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (70.4 - 59.0i)T + (1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (16.7 + 28.9i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.36 + 3.04i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.85334583378358011363528916825, −11.18352676502253378092365618521, −10.33707539050090152992832595902, −9.769974959913613326558875953998, −8.410776890488138848692161371265, −7.16284817763761619417834030962, −6.37597488394516396578657812843, −5.39161391106747743437713387374, −4.02343494499251887558767892424, −1.59562815584170447956623147125,
1.10046386723032101785586623361, 1.74648660826013383038971326011, 4.33876048364885749644877290622, 5.30100461567377794207272136710, 6.71859413787138766214794162634, 8.221464095933153055642081729345, 8.668856669051683135212257649655, 9.748855122125776745567950314953, 11.17517703372354853045958957205, 11.74129093289570667452193772796
