| L(s) = 1 | + (0.0941 + 1.41i)2-s + (0.553 − 2.78i)3-s + (−1.98 + 0.265i)4-s + (2.59 + 1.73i)5-s + (3.98 + 0.519i)6-s + (−1.96 + 0.813i)7-s + (−0.561 − 2.77i)8-s + (−4.67 − 1.93i)9-s + (−2.20 + 3.82i)10-s + (−2.02 + 0.402i)11-s + (−0.358 + 5.66i)12-s + (−1.99 + 1.33i)13-s + (−1.33 − 2.69i)14-s + (6.27 − 6.27i)15-s + (3.85 − 1.05i)16-s + (−2.31 − 2.31i)17-s + ⋯ |
| L(s) = 1 | + (0.0665 + 0.997i)2-s + (0.319 − 1.60i)3-s + (−0.991 + 0.132i)4-s + (1.16 + 0.776i)5-s + (1.62 + 0.212i)6-s + (−0.742 + 0.307i)7-s + (−0.198 − 0.980i)8-s + (−1.55 − 0.645i)9-s + (−0.697 + 1.21i)10-s + (−0.610 + 0.121i)11-s + (−0.103 + 1.63i)12-s + (−0.553 + 0.369i)13-s + (−0.356 − 0.720i)14-s + (1.61 − 1.61i)15-s + (0.964 − 0.263i)16-s + (−0.560 − 0.560i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.951325 + 0.144546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951325 + 0.144546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0941 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.553 + 2.78i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-2.59 - 1.73i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (1.96 - 0.813i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (2.02 - 0.402i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.99 - 1.33i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (2.31 + 2.31i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.37 - 3.55i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.993 + 2.39i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-2.34 - 0.465i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 1.81iT - 31T^{2} \) |
| 37 | \( 1 + (-1.40 + 2.10i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-4.66 + 11.2i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.452 - 2.27i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (2.27 + 2.27i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.98 + 1.58i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-7.08 - 4.73i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (1.95 - 9.85i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-2.88 + 14.5i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (6.41 - 2.65i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (3.36 + 1.39i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (10.9 - 10.9i)T - 79iT^{2} \) |
| 83 | \( 1 + (3.72 + 5.57i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (3.10 + 7.49i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 8.49iT - 97T^{2} \) |
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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
−14.60567015988161751695059086265, −13.90011371359959569541237765587, −13.15131612864655675998333299191, −12.25090744530604549286861613645, −10.09557910869076510150527529324, −8.906434637043632169608884140958, −7.42495361795732645034861384350, −6.66365608258751947233422052008, −5.70057207086243110619646391566, −2.59264926806811437568136031799,
2.90474528779099946316428149210, 4.52415553177435576722209162203, 5.55077250165240194182667676575, 8.603247218781860017442237726478, 9.681518259524564250974186935109, 9.967717407978693742908690669469, 11.16759978161865329308588879032, 12.91929831775246550358997496025, 13.53971613852723341281661616561, 14.79886162525923138894765335671
