L(s) = 1 | + (1.67 + 0.441i)3-s + (−1.40 − 2.43i)5-s + (2.60 + 1.47i)9-s + (4.74 + 2.74i)11-s + 1.35i·13-s + (−1.27 − 4.69i)15-s + (2.88 − 5.00i)17-s + (−1.71 + 0.992i)19-s + (−2.09 + 1.21i)23-s + (−1.44 + 2.49i)25-s + (3.71 + 3.63i)27-s − 7.05i·29-s + (3.07 + 1.77i)31-s + (6.73 + 6.68i)33-s + (−2.14 − 3.71i)37-s + ⋯ |
L(s) = 1 | + (0.966 + 0.254i)3-s + (−0.627 − 1.08i)5-s + (0.869 + 0.493i)9-s + (1.43 + 0.826i)11-s + 0.376i·13-s + (−0.329 − 1.21i)15-s + (0.700 − 1.21i)17-s + (−0.394 + 0.227i)19-s + (−0.437 + 0.252i)23-s + (−0.288 + 0.499i)25-s + (0.715 + 0.698i)27-s − 1.31i·29-s + (0.552 + 0.318i)31-s + (1.17 + 1.16i)33-s + (−0.352 − 0.610i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.300670689\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.300670689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.67 - 0.441i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.40 + 2.43i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.74 - 2.74i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 + (-2.88 + 5.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.71 - 0.992i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.09 - 1.21i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.05iT - 29T^{2} \) |
| 31 | \( 1 + (-3.07 - 1.77i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.14 + 3.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + (0.201 + 0.348i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.28 - 3.04i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.28 + 2.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.75 + 2.74i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.45 + 5.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 + (-0.295 - 0.170i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 - 2.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + (0.576 + 0.998i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−9.390052239835367797909833480051, −9.100090340155417279158191006474, −8.138657489409752714039107591742, −7.50335306328610977800141838660, −6.58742263087508829987391876906, −5.18515834954607438623544959842, −4.25417715099514097960737275873, −3.85799645691574157949090366062, −2.36873079407599770481568831762, −1.12624595128882760231622091997,
1.31018323868598208475292117747, 2.74963962992301320191416509826, 3.58079487099452931560343545045, 4.12424693682075104635116709689, 5.87280186049681350223123875599, 6.69435043241236679749025563104, 7.31099365958506792602056324012, 8.320546424127588301016335844833, 8.719068954394131603766151206625, 9.825612073415346461538805450808