| L(s) = 1 | + (0.255 − 2.26i)2-s + (0.238 − 1.71i)3-s + (−3.13 − 0.715i)4-s + (1.22 + 1.53i)5-s + (−3.83 − 0.980i)6-s + (0.968 + 4.24i)7-s + (−0.917 + 2.62i)8-s + (−2.88 − 0.819i)9-s + (3.79 − 2.38i)10-s + (−0.304 − 0.870i)11-s + (−1.97 + 5.20i)12-s + (−0.526 + 1.09i)13-s + (9.87 − 1.11i)14-s + (2.92 − 1.73i)15-s + (−0.0799 − 0.0385i)16-s + (−4.38 − 4.38i)17-s + ⋯ |
| L(s) = 1 | + (0.180 − 1.60i)2-s + (0.137 − 0.990i)3-s + (−1.56 − 0.357i)4-s + (0.546 + 0.685i)5-s + (−1.56 − 0.400i)6-s + (0.366 + 1.60i)7-s + (−0.324 + 0.926i)8-s + (−0.962 − 0.273i)9-s + (1.19 − 0.753i)10-s + (−0.0918 − 0.262i)11-s + (−0.570 + 1.50i)12-s + (−0.146 + 0.303i)13-s + (2.64 − 0.297i)14-s + (0.754 − 0.447i)15-s + (−0.0199 − 0.00962i)16-s + (−1.06 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.470271 - 0.987849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.470271 - 0.987849i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.238 + 1.71i)T \) |
| 29 | \( 1 + (4.21 - 3.35i)T \) |
| good | 2 | \( 1 + (-0.255 + 2.26i)T + (-1.94 - 0.445i)T^{2} \) |
| 5 | \( 1 + (-1.22 - 1.53i)T + (-1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.968 - 4.24i)T + (-6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (0.304 + 0.870i)T + (-8.60 + 6.85i)T^{2} \) |
| 13 | \( 1 + (0.526 - 1.09i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (4.38 + 4.38i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.885 - 1.40i)T + (-8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 1.92i)T + (5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-10.4 - 1.18i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (5.88 + 2.05i)T + (28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-2.44 + 2.44i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.647 + 5.74i)T + (-41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (2.99 - 1.04i)T + (36.7 - 29.3i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 1.36i)T + (11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 4.23iT - 59T^{2} \) |
| 61 | \( 1 + (3.53 + 2.22i)T + (26.4 + 54.9i)T^{2} \) |
| 67 | \( 1 + (-0.236 - 0.491i)T + (-41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (-6.12 - 2.94i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-8.47 + 0.955i)T + (71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (3.08 - 8.80i)T + (-61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (4.18 + 0.954i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (0.344 - 3.05i)T + (-86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-4.21 + 2.64i)T + (42.0 - 87.3i)T^{2} \) |
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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
−13.60045884026298975281162665856, −12.47104819549701312803390582353, −11.75923960788526984137185649988, −10.98776385953030720155616867341, −9.496264936181301872720678068535, −8.609825055792188119128677839616, −6.74621906942629744077859533345, −5.28724276910908227971851166618, −2.89319238891776660226890106174, −2.07194348403665792029171121674,
4.23850047142526075239322964854, 4.98887988013924466138304260197, 6.41073636378853911480984468740, 7.78007487714760605757568785490, 8.760743213531379105257865738588, 9.954350866222348502043664214729, 11.03749103360124355481617780841, 13.20137930094618903667526429994, 13.74681557728426702931960781304, 14.82427602368752049261617172722
