L(s) = 1 | + (−2.88 + 0.833i)3-s − 7.26i·5-s + (3.75 − 6.50i)7-s + (7.61 − 4.80i)9-s + (−14.6 − 8.48i)11-s + (10.1 − 17.5i)13-s + (6.05 + 20.9i)15-s + (12.9 + 7.46i)17-s + (−17.9 − 6.31i)19-s + (−5.40 + 21.8i)21-s + (28.8 + 16.6i)23-s − 27.8·25-s + (−17.9 + 20.1i)27-s + 30.3i·29-s + (−23.8 − 41.3i)31-s + ⋯ |
L(s) = 1 | + (−0.960 + 0.277i)3-s − 1.45i·5-s + (0.536 − 0.929i)7-s + (0.845 − 0.533i)9-s + (−1.33 − 0.771i)11-s + (0.780 − 1.35i)13-s + (0.403 + 1.39i)15-s + (0.760 + 0.439i)17-s + (−0.943 − 0.332i)19-s + (−0.257 + 1.04i)21-s + (1.25 + 0.723i)23-s − 1.11·25-s + (−0.664 + 0.747i)27-s + 1.04i·29-s + (−0.769 − 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9785171621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9785171621\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.88 - 0.833i)T \) |
| 19 | \( 1 + (17.9 + 6.31i)T \) |
good | 5 | \( 1 + 7.26iT - 25T^{2} \) |
| 7 | \( 1 + (-3.75 + 6.50i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (14.6 + 8.48i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-10.1 + 17.5i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-12.9 - 7.46i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-28.8 - 16.6i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 30.3iT - 841T^{2} \) |
| 31 | \( 1 + (23.8 + 41.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 25.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.04iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (24.8 + 43.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 62.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-47.5 + 27.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 73.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 95.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-0.588 + 1.01i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (12.3 + 7.13i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (59.5 - 103. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.0750 - 0.130i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (41.9 + 24.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-67.4 + 38.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (47.2 + 81.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.14321559302665434227564880642, −8.960960830873348322079466782917, −8.126228041276737473931667576692, −7.41956792802319230211821501140, −5.85953145297554020944741047713, −5.35362563033088255589540871747, −4.56667865908898452736253910883, −3.45567608382087021554163670204, −1.23285538446620382723740052548, −0.44384189611553870991993115575,
1.83910312819815202251684930859, 2.79686171061327977220382574770, 4.41422485472577860713706566466, 5.37824598409918825926588200113, 6.31316284324455677269256588274, 7.00854057098121036118465501102, 7.81131314428980105921387784310, 8.966950469877293946898930138124, 10.23332758123913136054534144688, 10.67905175564444608714427385329