| L(s) = 1 | + (−1.38 − 0.282i)2-s + (0.297 − 1.70i)3-s + (1.84 + 0.783i)4-s + (−1.06 + 1.98i)5-s + (−0.894 + 2.28i)6-s + (1.02 + 0.204i)7-s + (−2.32 − 1.60i)8-s + (−2.82 − 1.01i)9-s + (2.03 − 2.45i)10-s + (3.73 + 4.55i)11-s + (1.88 − 2.90i)12-s + (1.37 + 2.56i)13-s + (−1.36 − 0.573i)14-s + (3.07 + 2.40i)15-s + (2.77 + 2.88i)16-s + (−0.811 + 0.336i)17-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.199i)2-s + (0.171 − 0.985i)3-s + (0.920 + 0.391i)4-s + (−0.474 + 0.888i)5-s + (−0.365 + 0.930i)6-s + (0.388 + 0.0771i)7-s + (−0.823 − 0.567i)8-s + (−0.941 − 0.338i)9-s + (0.642 − 0.775i)10-s + (1.12 + 1.37i)11-s + (0.543 − 0.839i)12-s + (0.380 + 0.711i)13-s + (−0.364 − 0.153i)14-s + (0.793 + 0.620i)15-s + (0.693 + 0.720i)16-s + (−0.196 + 0.0815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.883358 + 0.104593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.883358 + 0.104593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.282i)T \) |
| 3 | \( 1 + (-0.297 + 1.70i)T \) |
| good | 5 | \( 1 + (1.06 - 1.98i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-1.02 - 0.204i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-3.73 - 4.55i)T + (-2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 2.56i)T + (-7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (0.811 - 0.336i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (3.88 + 1.17i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-5.89 - 3.93i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-4.65 + 5.66i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.43 - 1.43i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.85 + 1.47i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (5.92 + 3.96i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-8.34 - 0.822i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-5.52 + 2.28i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-7.96 - 9.70i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (4.52 - 8.46i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-4.37 + 0.430i)T + (59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (0.598 + 6.07i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (12.4 + 2.47i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (0.913 + 4.59i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (3.71 - 8.95i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (5.51 + 1.67i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (3.19 + 4.77i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.81 + 4.81i)T + 97iT^{2} \) |
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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
−11.44926113975779740307737365477, −10.59449866324389159626804397957, −9.303401962949068623831799008339, −8.676099580303447292895382290881, −7.41525378713308219456076641558, −7.05143698945368599542662358695, −6.21440825089369623356716540127, −4.08290159916844669088500432902, −2.64163829040526991126670157876, −1.51134166403497845753745495691,
0.892658070320736685245036938387, 3.04982247778663733739499050999, 4.34693254595337553570618072361, 5.52152821597133277364938408029, 6.57896954744655259037189359910, 8.188741858893520144112603636193, 8.593488933848777195381477600606, 9.158445062021989687907379880317, 10.42506448835096657566065445751, 11.05879158951575383841855489556
