| L(s) = 1 | + (0.955 − 1.04i)2-s + (1.59 + 0.669i)3-s + (−0.174 − 1.99i)4-s + (−3.00 − 1.09i)5-s + (2.22 − 1.02i)6-s + (4.58 + 0.808i)7-s + (−2.24 − 1.72i)8-s + (2.10 + 2.13i)9-s + (−4.01 + 2.09i)10-s + (−0.303 − 0.833i)11-s + (1.05 − 3.29i)12-s + (−1.22 − 1.45i)13-s + (5.22 − 4.00i)14-s + (−4.07 − 3.76i)15-s + (−3.93 + 0.693i)16-s + (−4.03 + 2.32i)17-s + ⋯ |
| L(s) = 1 | + (0.675 − 0.737i)2-s + (0.922 + 0.386i)3-s + (−0.0870 − 0.996i)4-s + (−1.34 − 0.489i)5-s + (0.908 − 0.418i)6-s + (1.73 + 0.305i)7-s + (−0.793 − 0.608i)8-s + (0.700 + 0.713i)9-s + (−1.27 + 0.661i)10-s + (−0.0914 − 0.251i)11-s + (0.304 − 0.952i)12-s + (−0.338 − 0.403i)13-s + (1.39 − 1.07i)14-s + (−1.05 − 0.972i)15-s + (−0.984 + 0.173i)16-s + (−0.978 + 0.564i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72232 - 0.976077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72232 - 0.976077i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.955 + 1.04i)T \) |
| 3 | \( 1 + (-1.59 - 0.669i)T \) |
| good | 5 | \( 1 + (3.00 + 1.09i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.58 - 0.808i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.303 + 0.833i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.22 + 1.45i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.03 - 2.32i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.171 - 0.296i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 5.68i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.16 - 3.49i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (7.80 - 1.37i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.31 + 1.33i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0346 - 0.0413i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.85 - 1.04i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.90 + 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + (-2.43 + 6.68i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.19 + 1.44i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.31 - 5.29i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.186 - 0.323i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.29 + 10.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.54 + 3.03i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.90 + 10.6i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.35 + 3.66i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.833 - 0.303i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.08588092250327519226635154259, −11.25942847761374271053601311650, −10.59813607947713887215606226195, −9.052126806907059034132083888135, −8.350736513247991589473520936128, −7.43555890469715314086028874397, −5.20952270915830964806678771770, −4.48863628013816114027280998637, −3.47699645058392041099103779251, −1.82520523264456277485650635454,
2.52185796971183232323530413213, 4.09683756501267269291674611115, 4.68490831800721314221217523595, 6.74984686228626289487601558755, 7.54518418922190080096579279204, 8.076633008042825680690234344755, 8.982234968975571025373938201860, 10.94410204922465933005616225925, 11.69653482355253617350597919759, 12.54306904729824359740997865102
