| L(s) = 1 | + (−0.257 − 2.45i)2-s + (−1.08 + 1.20i)3-s + (−3.99 + 0.849i)4-s + (−3.16 + 1.41i)5-s + (3.22 + 2.34i)6-s + (1.58 + 4.89i)8-s + (0.0399 + 0.379i)9-s + (4.27 + 7.40i)10-s + (−0.0978 − 3.31i)11-s + (3.30 − 5.72i)12-s + (0.528 − 0.384i)13-s + (1.73 − 5.33i)15-s + (4.12 − 1.83i)16-s + (0.118 − 1.13i)17-s + (0.921 − 0.195i)18-s + (5.94 + 1.26i)19-s + ⋯ |
| L(s) = 1 | + (−0.182 − 1.73i)2-s + (−0.625 + 0.694i)3-s + (−1.99 + 0.424i)4-s + (−1.41 + 0.630i)5-s + (1.31 + 0.957i)6-s + (0.561 + 1.72i)8-s + (0.0133 + 0.126i)9-s + (1.35 + 2.34i)10-s + (−0.0295 − 0.999i)11-s + (0.953 − 1.65i)12-s + (0.146 − 0.106i)13-s + (0.447 − 1.37i)15-s + (1.03 − 0.459i)16-s + (0.0288 − 0.274i)17-s + (0.217 − 0.0461i)18-s + (1.36 + 0.289i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.413862 - 0.467236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413862 - 0.467236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.0978 + 3.31i)T \) |
| good | 2 | \( 1 + (0.257 + 2.45i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (1.08 - 1.20i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (3.16 - 1.41i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.528 + 0.384i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.118 + 1.13i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-5.94 - 1.26i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.33 + 5.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.41 - 4.34i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.55 - 1.13i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (0.294 + 0.326i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (1.82 + 5.61i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (-0.591 - 0.125i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-8.97 - 3.99i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.65 - 0.352i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.26 + 2.78i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 5.35i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.38 - 3.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.55 - 1.39i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (0.277 + 2.63i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.41 + 3.93i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.349 + 0.604i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 8.73i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.72090747483031675553100901298, −10.27415398252482745890575966966, −9.086146722743098318439146255252, −8.239219593686643633592955912408, −7.16075568115086851004227739560, −5.49204217535833084611363020456, −4.42541508018696749863789507091, −3.59822009935029156678769969367, −2.80370044323849483259973072624, −0.66124668283747731145531326143,
0.862034029392637816277542909909, 3.83278883577116802137433004121, 4.83907858435809079539140561487, 5.67333862304749262986839419168, 6.78034881916322962238298949282, 7.45898991608072155165612010990, 7.87301599245351137803363055743, 8.981712428781932043242881026444, 9.712639562451127819956604115776, 11.43114442381874035573624730915
