| L(s) = 1 | + (−1.30 − 0.548i)2-s + (1.29 − 1.15i)3-s + (1.39 + 1.42i)4-s + (−1.03 − 0.508i)5-s + (−2.31 + 0.789i)6-s + (1.75 + 2.28i)7-s + (−1.04 − 2.63i)8-s + (0.353 − 2.97i)9-s + (1.06 + 1.22i)10-s + (−3.86 − 3.38i)11-s + (3.45 + 0.241i)12-s + (−5.56 − 0.364i)13-s + (−1.03 − 3.93i)14-s + (−1.91 + 0.527i)15-s + (−0.0850 + 3.99i)16-s + (0.422 + 0.422i)17-s + ⋯ |
| L(s) = 1 | + (−0.921 − 0.387i)2-s + (0.747 − 0.664i)3-s + (0.699 + 0.714i)4-s + (−0.460 − 0.227i)5-s + (−0.946 + 0.322i)6-s + (0.662 + 0.863i)7-s + (−0.367 − 0.929i)8-s + (0.117 − 0.993i)9-s + (0.336 + 0.388i)10-s + (−1.16 − 1.02i)11-s + (0.997 + 0.0695i)12-s + (−1.54 − 0.101i)13-s + (−0.276 − 1.05i)14-s + (−0.495 + 0.136i)15-s + (−0.0212 + 0.999i)16-s + (0.102 + 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.362016 - 0.831877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362016 - 0.831877i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.548i)T \) |
| 3 | \( 1 + (-1.29 + 1.15i)T \) |
| good | 5 | \( 1 + (1.03 + 0.508i)T + (3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (-1.75 - 2.28i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (3.86 + 3.38i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (5.56 + 0.364i)T + (12.8 + 1.69i)T^{2} \) |
| 17 | \( 1 + (-0.422 - 0.422i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.18 + 6.25i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.49 + 3.24i)T + (-5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 5.11i)T + (-23.0 - 17.6i)T^{2} \) |
| 31 | \( 1 + (-0.836 - 1.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.04 + 4.03i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-4.08 - 3.13i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (3.22 + 2.82i)T + (5.61 + 42.6i)T^{2} \) |
| 47 | \( 1 + (-2.07 - 7.74i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.33 + 0.465i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (3.59 - 7.29i)T + (-35.9 - 46.8i)T^{2} \) |
| 61 | \( 1 + (3.03 - 8.94i)T + (-48.3 - 37.1i)T^{2} \) |
| 67 | \( 1 + (7.78 - 6.82i)T + (8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (5.65 + 2.34i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-12.6 + 5.24i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (2.18 + 8.16i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (4.43 - 2.18i)T + (50.5 - 65.8i)T^{2} \) |
| 89 | \( 1 + (10.2 + 4.26i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-15.8 - 9.13i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.28838292992330229303428472010, −9.258948334676104811267049329534, −8.634956096026051942669713087705, −7.80035704648815364893964925953, −7.42160977196269203762756076907, −6.01882952592821316064732433153, −4.65860770812843347434456766800, −2.85685752381807489379277678438, −2.47428820817595636836303689195, −0.61144542986082515181380911891,
1.85028372837474664706962336919, 3.14262265123356376916687614455, 4.63779970606418598272060556614, 5.33348636324979791302453755778, 7.29820973021102011209990281409, 7.53777900501936226778461203635, 8.151244827554398696570470164313, 9.578509444131637456023115452322, 9.916538601293585230302581850249, 10.68269006860194297599127217691
