L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + 3·5-s + (−1.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 3·8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)10-s + (−2.5 + 4.33i)11-s + 1.73i·12-s + (−1 + 1.73i)13-s − 0.999·14-s + (−4.5 + 2.59i)15-s + (0.500 + 0.866i)16-s + (2.5 − 4.33i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (0.250 − 0.433i)4-s + 1.34·5-s + (−0.612 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s + (0.474 + 0.821i)10-s + (−0.753 + 1.30i)11-s + 0.499i·12-s + (−0.277 + 0.480i)13-s − 0.267·14-s + (−1.16 + 0.670i)15-s + (0.125 + 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20037 + 0.633002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20037 + 0.633002i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.94557418304685453704665704486, −12.04629196587565280467183349285, −10.55129676400131601881591675975, −10.13868505007393307386861931920, −9.207658597561480464347075941016, −7.16674136269042478493767604865, −6.38574148858297561264300594440, −5.36405632162187088604720244345, −4.72933796398282718705698572119, −2.10251270516430367850184925394,
1.70622857617598332888309681540, 3.23968468519268185190959903964, 5.15664700952504823783882866985, 6.03159128952584862550636425224, 7.21270551016907254918314328735, 8.424272868385385424520672541082, 10.18966887567098193450825032113, 10.67139776607494661805316815598, 11.61409407938119411404578755455, 12.88175615714191337441196109689