L(s) = 1 | + (0.537 − 0.310i)3-s + (−0.0337 + 0.0585i)5-s + (1.98 + 1.74i)7-s + (−1.30 + 2.26i)9-s + (1.28 + 2.22i)11-s + 5.63·13-s + 0.0419i·15-s + (−4.45 + 2.57i)17-s + (−5.90 − 3.40i)19-s + (1.60 + 0.323i)21-s + (−4.92 − 2.84i)23-s + (2.49 + 4.32i)25-s + 3.48i·27-s + 2.76i·29-s + (1.14 + 1.97i)31-s + ⋯ |
L(s) = 1 | + (0.310 − 0.179i)3-s + (−0.0151 + 0.0261i)5-s + (0.750 + 0.661i)7-s + (−0.435 + 0.754i)9-s + (0.387 + 0.671i)11-s + 1.56·13-s + 0.0108i·15-s + (−1.08 + 0.623i)17-s + (−1.35 − 0.781i)19-s + (0.351 + 0.0706i)21-s + (−1.02 − 0.593i)23-s + (0.499 + 0.865i)25-s + 0.670i·27-s + 0.513i·29-s + (0.205 + 0.355i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56869 + 0.840441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56869 + 0.840441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.98 - 1.74i)T \) |
good | 3 | \( 1 + (-0.537 + 0.310i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.0337 - 0.0585i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.28 - 2.22i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + (4.45 - 2.57i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.90 + 3.40i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.92 + 2.84i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.76iT - 29T^{2} \) |
| 31 | \( 1 + (-1.14 - 1.97i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.82 - 3.93i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.39iT - 41T^{2} \) |
| 43 | \( 1 - 6.99T + 43T^{2} \) |
| 47 | \( 1 + (-2.32 + 4.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.48 + 4.32i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.38 + 2.53i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.34 + 5.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.72 + 9.91i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.52iT - 71T^{2} \) |
| 73 | \( 1 + (0.767 - 0.442i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.07 + 1.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + (-1.73 - 1.00i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.52iT - 97T^{2} \) |
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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
−10.47582649312267794446612251998, −9.023150935288739510745831866609, −8.616205018942901723450193064633, −7.976164533689673334704612001406, −6.74710742829181268083087869160, −6.01022908062135860058027971700, −4.86183545754733071710369971601, −4.03388457867413324366468416923, −2.52935733051762649586860608389, −1.68938197410722392422834822575,
0.867341025004569061969826493750, 2.40103698512760305613027160273, 3.96810091454759955355011354583, 4.13407741086994579340130827592, 5.89354323801744552956786675176, 6.32810134806734662270674321359, 7.58270193576656631861342453632, 8.612831155395517247818515852636, 8.784947129899594854121057615738, 10.03162761471511935762572345906