| L(s) = 1 | + (1.34 − 1.09i)3-s + (0.181 − 1.03i)5-s + (0.246 − 0.206i)7-s + (0.591 − 2.94i)9-s + (−0.447 − 2.53i)11-s + (−1.61 + 0.588i)13-s + (−0.888 − 1.58i)15-s + (0.747 − 1.29i)17-s + (−1.58 − 2.73i)19-s + (0.103 − 0.547i)21-s + (−1.68 − 1.41i)23-s + (3.66 + 1.33i)25-s + (−2.43 − 4.59i)27-s + (1.03 + 0.377i)29-s + (1.18 + 0.990i)31-s + ⋯ |
| L(s) = 1 | + (0.773 − 0.633i)3-s + (0.0813 − 0.461i)5-s + (0.0931 − 0.0781i)7-s + (0.197 − 0.980i)9-s + (−0.134 − 0.765i)11-s + (−0.448 + 0.163i)13-s + (−0.229 − 0.408i)15-s + (0.181 − 0.314i)17-s + (−0.362 − 0.628i)19-s + (0.0225 − 0.119i)21-s + (−0.351 − 0.294i)23-s + (0.733 + 0.266i)25-s + (−0.468 − 0.883i)27-s + (0.192 + 0.0700i)29-s + (0.212 + 0.177i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24042 - 1.44554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24042 - 1.44554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.34 + 1.09i)T \) |
| good | 5 | \( 1 + (-0.181 + 1.03i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.246 + 0.206i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.447 + 2.53i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.61 - 0.588i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.747 + 1.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.68 + 1.41i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 0.377i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 0.990i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.26 - 5.65i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.73 + 1.35i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.52 + 8.66i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.47 + 3.75i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 7.88T + 53T^{2} \) |
| 59 | \( 1 + (-1.63 + 9.27i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.77 - 5.68i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.44 + 2.34i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.30 - 10.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.92 - 5.07i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.69 + 1.70i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.703 - 0.255i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.09 - 3.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.759 - 4.30i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.770571280910459177162712266391, −8.818649649682700997105018697092, −8.449278716801001806444061098579, −7.38519490062231377120808544552, −6.68203200016973494843623554618, −5.57439848005919580629938184454, −4.48408745378496487715099606618, −3.29191541861737321059975595445, −2.27520629097441238477979002194, −0.854746332224842229044456162728,
1.96346163086102701338169096420, 2.94066961771232433106688025303, 4.03811967382915493530762800335, 4.89730909215516071332936193952, 6.00842012913901342330607378665, 7.20021757766901102390346958577, 7.88353613456847117980833944340, 8.762648969185899043286236282760, 9.661136854410993420778191321749, 10.27906052004972034369403136736
