| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (2.05 − 1.32i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.347 + 2.41i)10-s + (−1.49 − 3.27i)11-s + (0.415 + 0.909i)12-s + (−0.524 − 3.64i)13-s + (0.841 + 0.540i)14-s + (−2.34 + 0.687i)15-s + (−0.142 + 0.989i)16-s + (−1.92 + 2.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.918 − 0.590i)5-s + (0.267 − 0.308i)6-s + (0.0537 − 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.109 + 0.764i)10-s + (−0.450 − 0.987i)11-s + (0.119 + 0.262i)12-s + (−0.145 − 1.01i)13-s + (0.224 + 0.144i)14-s + (−0.604 + 0.177i)15-s + (−0.0355 + 0.247i)16-s + (−0.467 + 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.546270 - 0.631068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546270 - 0.631068i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-2.10 + 4.30i)T \) |
| good | 5 | \( 1 + (-2.05 + 1.32i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.49 + 3.27i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.524 + 3.64i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.92 - 2.22i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (0.293 + 0.339i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.42 - 6.25i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.71 - 1.97i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (1.58 + 1.02i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.41 + 0.910i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (7.74 + 2.27i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 + (-1.11 + 7.78i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.48 + 10.3i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.352 + 0.103i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (0.186 - 0.407i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-5.82 + 12.7i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.33 - 8.46i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.53 + 10.6i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.0437 - 0.0280i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-14.9 - 4.39i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (9.45 - 6.07i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.722442633727004263582846666189, −8.854062957045251734672425748041, −8.182021621934115025952658395414, −7.17296367570021965581899467287, −6.30327555864099008789870098034, −5.43734667343093833532747296403, −5.04552177108752737000616564196, −3.50946805912034173219620884265, −1.81330435141558981599480307430, −0.44328875535502073372691593738,
1.79232484607443920355793892760, 2.52490014259856451451172373953, 3.98307603755963039334978779965, 4.99805700425270093327333308991, 5.88711817169269770369734938871, 6.89193844396841221884994723712, 7.61782259649350955359101832398, 9.041425189843644754039074250227, 9.583901620296467104180698395338, 10.15059349040122650186111457444
