| L(s) = 1 | + (0.241 + 0.0513i)2-s + (0.762 − 2.34i)3-s + (−1.77 − 0.788i)4-s + (−1.68 + 2.91i)5-s + (0.304 − 0.527i)6-s + (0.00261 + 0.0248i)7-s + (−0.787 − 0.572i)8-s + (−2.49 − 1.81i)9-s + (−0.555 + 0.617i)10-s + (−5.85 − 2.60i)11-s + (−3.20 + 3.55i)12-s + (−2.85 + 2.20i)13-s + (−0.000646 + 0.00615i)14-s + (5.55 + 6.16i)15-s + (2.43 + 2.70i)16-s + (2.24 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.170 + 0.0363i)2-s + (0.440 − 1.35i)3-s + (−0.885 − 0.394i)4-s + (−0.751 + 1.30i)5-s + (0.124 − 0.215i)6-s + (0.000989 + 0.00941i)7-s + (−0.278 − 0.202i)8-s + (−0.832 − 0.605i)9-s + (−0.175 + 0.195i)10-s + (−1.76 − 0.785i)11-s + (−0.924 + 1.02i)12-s + (−0.791 + 0.610i)13-s + (−0.000172 + 0.00164i)14-s + (1.43 + 1.59i)15-s + (0.608 + 0.675i)16-s + (0.544 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0443880 + 0.208230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0443880 + 0.208230i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.85 - 2.20i)T \) |
| 31 | \( 1 + (0.730 + 5.51i)T \) |
| good | 2 | \( 1 + (-0.241 - 0.0513i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.762 + 2.34i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (1.68 - 2.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.00261 - 0.0248i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (5.85 + 2.60i)T + (7.36 + 8.17i)T^{2} \) |
| 17 | \( 1 + (-2.24 + 1.00i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.67 + 0.993i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.04 - 0.910i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-7.08 - 1.50i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 + (5.02 - 5.57i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (10.3 + 2.19i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.13 + 6.55i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.07 - 0.923i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.950 - 1.05i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (3.57 + 6.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.08 + 12.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.471 - 0.342i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.912 - 8.68i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.761 + 7.24i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.01 - 0.853i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-1.66 - 15.7i)T + (-87.0 + 18.5i)T^{2} \) |
| 97 | \( 1 + (4.93 + 2.19i)T + (64.9 + 72.0i)T^{2} \) |
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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
−10.70105184429618454016503772635, −10.01288145175641293918082433759, −8.486550933278703849912440102742, −7.942772137035985844783999925360, −7.07245785376648956983352055403, −6.19355073626241446385274369097, −4.88796540025986612678004443157, −3.36069095610174393541271207195, −2.35693694574951279668170869796, −0.12056027799717403017013165458,
2.93534782944738049744964567967, 4.13155287874511273241285103608, 4.81154267396202823272373166716, 5.27860138317690817226570830728, 7.66655873695242781755345914162, 8.345152253605734964097413188874, 8.899771720199902453089749966478, 10.16861684107748522606374363517, 10.28866730704855200661568378485, 12.15207729222552920414189625601
