| L(s) = 1 | + (−1.01 + 0.982i)2-s + (1.51 + 0.301i)3-s + (0.0705 − 1.99i)4-s + (1.90 − 1.27i)5-s + (−1.83 + 1.18i)6-s + (−2.53 + 3.79i)7-s + (1.89 + 2.10i)8-s + (−0.571 − 0.236i)9-s + (−0.689 + 3.17i)10-s + (−0.737 − 3.70i)11-s + (0.708 − 3.00i)12-s + (1.06 + 1.06i)13-s + (−1.14 − 6.34i)14-s + (3.27 − 1.35i)15-s + (−3.99 − 0.282i)16-s + (−3.21 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.719 + 0.694i)2-s + (0.873 + 0.173i)3-s + (0.0352 − 0.999i)4-s + (0.853 − 0.570i)5-s + (−0.749 + 0.481i)6-s + (−0.957 + 1.43i)7-s + (0.668 + 0.743i)8-s + (−0.190 − 0.0789i)9-s + (−0.218 + 1.00i)10-s + (−0.222 − 1.11i)11-s + (0.204 − 0.867i)12-s + (0.296 + 0.296i)13-s + (−0.306 − 1.69i)14-s + (0.845 − 0.350i)15-s + (−0.997 − 0.0705i)16-s + (−0.779 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.783262 + 0.294622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.783262 + 0.294622i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.01 - 0.982i)T \) |
| 17 | \( 1 + (3.21 + 2.58i)T \) |
| good | 3 | \( 1 + (-1.51 - 0.301i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.90 + 1.27i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (2.53 - 3.79i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.737 + 3.70i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 1.06i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.308 + 0.745i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.55 + 0.310i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 1.58i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.416 + 2.09i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (1.44 - 7.27i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.06 - 4.05i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.146 - 0.352i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.465 + 0.465i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.45 + 1.01i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.44 - 2.25i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (3.92 - 5.87i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + (7.36 + 1.46i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-5.84 + 3.90i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.33 - 11.7i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (6.25 - 2.58i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-5.52 + 5.52i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.39 - 5.07i)T + (-37.1 + 89.6i)T^{2} \) |
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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
−15.15602755843806298462023623060, −13.92996871455399447788309353191, −13.15784017080941074260902739306, −11.42147554574582289046471744816, −9.693593848062389524257271820775, −9.015712601775120761678046807818, −8.457059779558212008793689279349, −6.39435195782667405972823200368, −5.47175352167542833464378305312, −2.64062117840057742364345377636,
2.33133479002557189803400117521, 3.79822225264583379940443334894, 6.69400101880404687249979070462, 7.71789660855638907039435682246, 9.172648371318575948651031395260, 10.14012947285744422669486568101, 10.81365029477986660760911497812, 12.74826016291041813357225824269, 13.42269788038736604425815219282, 14.33031182266599449096949098373
