| L(s) = 1 | + (−0.296 − 2.98i)3-s + (3.94 − 5.43i)5-s + (−0.935 + 2.87i)7-s + (−8.82 + 1.76i)9-s + (−6.22 − 9.07i)11-s + (−0.130 + 0.0946i)13-s + (−17.3 − 10.1i)15-s + (14.3 − 19.7i)17-s + (−0.192 − 0.591i)19-s + (8.87 + 1.93i)21-s + 6.09i·23-s + (−6.20 − 19.0i)25-s + (7.89 + 25.8i)27-s + (49.6 + 16.1i)29-s + (−36.1 + 26.2i)31-s + ⋯ |
| L(s) = 1 | + (−0.0987 − 0.995i)3-s + (0.789 − 1.08i)5-s + (−0.133 + 0.411i)7-s + (−0.980 + 0.196i)9-s + (−0.565 − 0.824i)11-s + (−0.0100 + 0.00727i)13-s + (−1.15 − 0.678i)15-s + (0.844 − 1.16i)17-s + (−0.0101 − 0.0311i)19-s + (0.422 + 0.0923i)21-s + 0.264i·23-s + (−0.248 − 0.763i)25-s + (0.292 + 0.956i)27-s + (1.71 + 0.556i)29-s + (−1.16 + 0.846i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.840560 - 1.10020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840560 - 1.10020i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.296 + 2.98i)T \) |
| 11 | \( 1 + (6.22 + 9.07i)T \) |
| good | 5 | \( 1 + (-3.94 + 5.43i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (0.935 - 2.87i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (0.130 - 0.0946i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-14.3 + 19.7i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (0.192 + 0.591i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 - 6.09iT - 529T^{2} \) |
| 29 | \( 1 + (-49.6 - 16.1i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (36.1 - 26.2i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-9.51 + 29.2i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (1.80 - 0.585i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 79.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + (14.0 - 4.56i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-28.0 - 38.6i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (20.8 + 6.77i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-49.8 - 36.2i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 56.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-52.9 + 72.9i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (21.7 - 67.0i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-79.8 + 58.0i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (60.8 - 83.7i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 165. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (79.1 - 57.5i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−12.68685167891488160307992302315, −12.10327872608005785643970910746, −10.79856002134497587257120702781, −9.320655351604491658703826116379, −8.559787651350014237949117419103, −7.35757487962635547014293860825, −5.88166356105158184996872205082, −5.20204218924125664321511218846, −2.74089716873212650622260900913, −1.01777200930924035325154420164,
2.59680062002240595783735851813, 4.04230014897372180267666753136, 5.54674427870091436119981662593, 6.62458953907386190094190906097, 8.044078224046244674187177733569, 9.605285009357391168562561779203, 10.27741575810260521117610021233, 10.83996723354535522651844837303, 12.25597623841167465926833690300, 13.54387360379321922078795492717
