| L(s) = 1 | + (−1.41 − 0.0923i)2-s + (2.17 − 1.09i)3-s + (1.98 + 0.260i)4-s + (1.51 + 3.41i)5-s + (−3.16 + 1.34i)6-s + (0.637 + 0.382i)7-s + (−2.77 − 0.551i)8-s + (1.74 − 2.34i)9-s + (−1.81 − 4.95i)10-s + (−1.10 − 0.861i)11-s + (4.59 − 1.60i)12-s + (3.00 − 2.86i)13-s + (−0.864 − 0.598i)14-s + (7.01 + 5.75i)15-s + (3.86 + 1.03i)16-s + (−1.81 − 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0653i)2-s + (1.25 − 0.631i)3-s + (0.991 + 0.130i)4-s + (0.676 + 1.52i)5-s + (−1.29 + 0.548i)6-s + (0.240 + 0.144i)7-s + (−0.980 − 0.194i)8-s + (0.580 − 0.782i)9-s + (−0.575 − 1.56i)10-s + (−0.332 − 0.259i)11-s + (1.32 − 0.462i)12-s + (0.833 − 0.793i)13-s + (−0.231 − 0.159i)14-s + (1.81 + 1.48i)15-s + (0.966 + 0.258i)16-s + (−0.439 − 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60258 + 0.148477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60258 + 0.148477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0923i)T \) |
| good | 3 | \( 1 + (-2.17 + 1.09i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-1.51 - 3.41i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-0.637 - 0.382i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (1.10 + 0.861i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.00 + 2.86i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.81 + 2.21i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.716 - 4.13i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-2.54 - 1.20i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (5.64 + 3.19i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-6.24 + 1.24i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-3.68 - 2.33i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (2.10 + 2.32i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.31 - 7.00i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-3.87 - 2.06i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (9.93 - 5.62i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (3.81 - 4.01i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (1.01 + 13.7i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (9.44 + 8.14i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-6.84 - 1.01i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-5.66 + 3.39i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (7.45 + 2.26i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-10.0 + 6.39i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (8.45 - 4.00i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (11.1 + 7.46i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.81035523784952896640316421221, −9.884854373425016822302566059648, −9.161065351365434656365801730359, −8.021936343527471745221670669577, −7.69285995332553763301071279613, −6.61287064244500543939970586201, −5.86571859076735487822683880252, −3.30435503017494518321685153827, −2.75314816616870037104410626001, −1.70103738963527613086940486037,
1.37479447545529709565063489057, 2.51552867559420695779027213916, 4.05639176484158521284573602827, 5.13168525610825956017452016533, 6.42092852602431937083063543369, 7.73515486686317896330831988720, 8.656013360556446090948667928794, 8.968530261491588028662518263892, 9.564309821125757290396320842484, 10.51226871738603022068752565160
