| L(s) = 1 | + (−0.149 − 1.42i)2-s + (−2.44 − 4.23i)3-s + (1.91 − 0.406i)4-s + (−3.74 + 3.37i)5-s + (−5.65 + 4.10i)6-s + (0.725 + 6.96i)7-s + (−2.63 − 8.09i)8-s + (−7.44 + 12.8i)9-s + (5.35 + 4.82i)10-s + (13.4 + 12.0i)11-s + (−6.39 − 7.09i)12-s + (−15.0 + 10.9i)13-s + (9.79 − 2.07i)14-s + (23.4 + 7.60i)15-s + (−3.98 + 1.77i)16-s + (−13.4 + 14.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.0747 − 0.711i)2-s + (−0.814 − 1.41i)3-s + (0.478 − 0.101i)4-s + (−0.748 + 0.674i)5-s + (−0.942 + 0.684i)6-s + (0.103 + 0.994i)7-s + (−0.328 − 1.01i)8-s + (−0.826 + 1.43i)9-s + (0.535 + 0.482i)10-s + (1.21 + 1.09i)11-s + (−0.532 − 0.591i)12-s + (−1.15 + 0.838i)13-s + (0.699 − 0.148i)14-s + (1.56 + 0.507i)15-s + (−0.248 + 0.110i)16-s + (−0.793 + 0.880i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.642513 + 0.161958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642513 + 0.161958i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.725 - 6.96i)T \) |
| 41 | \( 1 + (-10.3 + 39.6i)T \) |
| good | 2 | \( 1 + (0.149 + 1.42i)T + (-3.91 + 0.831i)T^{2} \) |
| 3 | \( 1 + (2.44 + 4.23i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (3.74 - 3.37i)T + (2.61 - 24.8i)T^{2} \) |
| 11 | \( 1 + (-13.4 - 12.0i)T + (12.6 + 120. i)T^{2} \) |
| 13 | \( 1 + (15.0 - 10.9i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (13.4 - 14.9i)T + (-30.2 - 287. i)T^{2} \) |
| 19 | \( 1 + (4.12 - 1.83i)T + (241. - 268. i)T^{2} \) |
| 23 | \( 1 + (2.84 + 27.0i)T + (-517. + 109. i)T^{2} \) |
| 29 | \( 1 + (6.00 + 1.95i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-20.3 - 18.3i)T + (100. + 955. i)T^{2} \) |
| 37 | \( 1 + (8.85 + 9.83i)T + (-143. + 1.36e3i)T^{2} \) |
| 43 | \( 1 + (24.0 - 17.4i)T + (571. - 1.75e3i)T^{2} \) |
| 47 | \( 1 + (-7.28 - 69.2i)T + (-2.16e3 + 459. i)T^{2} \) |
| 53 | \( 1 + (-4.23 - 19.9i)T + (-2.56e3 + 1.14e3i)T^{2} \) |
| 59 | \( 1 + (-30.3 + 68.1i)T + (-2.32e3 - 2.58e3i)T^{2} \) |
| 61 | \( 1 + (-35.1 - 79.0i)T + (-2.48e3 + 2.76e3i)T^{2} \) |
| 67 | \( 1 + (-14.3 - 67.5i)T + (-4.10e3 + 1.82e3i)T^{2} \) |
| 71 | \( 1 + (86.9 - 28.2i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (92.6 - 53.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (31.4 + 18.1i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 28.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-78.6 + 35.0i)T + (5.30e3 - 5.88e3i)T^{2} \) |
| 97 | \( 1 + (19.5 - 60.1i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.72275992768448269980261373382, −11.25776534801936139585414058277, −10.09387507077258980434524553230, −8.813562441347634262555214809432, −7.33619308904578856729297095726, −6.82356242623633091029279308528, −6.11532152914298201487730044957, −4.33702148375058542280266359627, −2.48310580002009348987534210726, −1.67134130179829779163249245404,
0.35862965753919874328281310211, 3.43731871495169906905138330577, 4.48970203797726183693404848128, 5.35131880872516553956863069763, 6.50601812433046647251400377218, 7.58165871090191557542542085058, 8.623443472262437556732567558009, 9.671052524789586247136821356205, 10.67881055255937889413032564634, 11.66728356420159320317718523081
