| L(s) = 1 | + (0.654 − 0.755i)2-s + (−2.75 + 1.76i)3-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.465 + 3.23i)6-s + (0.267 − 0.0784i)7-s + (−0.841 − 0.540i)8-s + (3.20 − 7.00i)9-s + (−0.959 − 0.281i)10-s + (−3.85 − 4.44i)11-s + (2.14 + 2.47i)12-s + (−4.74 − 1.39i)13-s + (0.115 − 0.253i)14-s + (2.75 + 1.76i)15-s + (−0.959 + 0.281i)16-s + (0.328 − 2.28i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−1.58 + 1.02i)3-s + (−0.0711 − 0.494i)4-s + (−0.185 − 0.406i)5-s + (−0.190 + 1.32i)6-s + (0.100 − 0.0296i)7-s + (−0.297 − 0.191i)8-s + (1.06 − 2.33i)9-s + (−0.303 − 0.0890i)10-s + (−1.16 − 1.34i)11-s + (0.618 + 0.713i)12-s + (−1.31 − 0.386i)13-s + (0.0309 − 0.0676i)14-s + (0.710 + 0.456i)15-s + (−0.239 + 0.0704i)16-s + (0.0797 − 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.176411 - 0.437139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.176411 - 0.437139i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-2.27 - 4.22i)T \) |
| good | 3 | \( 1 + (2.75 - 1.76i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-0.267 + 0.0784i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.85 + 4.44i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (4.74 + 1.39i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.328 + 2.28i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.284 - 1.97i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 8.16i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.292 + 0.187i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 3.18i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.18 - 4.78i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.05 - 3.89i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 + (-1.39 + 0.410i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.86 + 1.42i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.61 - 2.96i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.87 + 7.93i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-7.64 + 8.82i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.0126 - 0.0879i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (4.51 + 1.32i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.67 + 8.04i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 7.53i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.97 + 4.31i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.60274700956716971246485405279, −11.12287565338299859835983559983, −10.12860061811449218974229939599, −9.499762939427076151450104330524, −7.82017361343307187002705216743, −6.15239958774906271436520771988, −5.28876939661226050177616561347, −4.68930476050794393731303134080, −3.25722966300733592451010755528, −0.38654777507997853657291210139,
2.26378846140654355094434888378, 4.76963829628599601408920720060, 5.25554882382671688360122561842, 6.76249857548769436084221078788, 7.06759908951565528599994420524, 8.052236234494577588292405748474, 10.03221841598820002410871040050, 10.86986354170258144242519854274, 11.90975651766351293138233766798, 12.60034919368870117318074684032
