| 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
−14.79886162525923138894765335671, −13.53971613852723341281661616561, −12.91929831775246550358997496025, −11.16759978161865329308588879032, −9.967717407978693742908690669469, −9.681518259524564250974186935109, −8.603247218781860017442237726478, −5.55077250165240194182667676575, −4.52415553177435576722209162203, −2.90474528779099946316428149210,
2.59264926806811437568136031799, 5.70057207086243110619646391566, 6.66365608258751947233422052008, 7.42495361795732645034861384350, 8.906434637043632169608884140958, 10.09557910869076510150527529324, 12.25090744530604549286861613645, 13.15131612864655675998333299191, 13.90011371359959569541237765587, 14.60567015988161751695059086265
