| L(s) = 1 | + (0.0891 + 1.41i)2-s + (3.05 + 1.26i)3-s + (−1.98 + 0.251i)4-s + (0.290 − 0.147i)5-s + (−1.51 + 4.43i)6-s + (−1.49 − 2.43i)7-s + (−0.532 − 2.77i)8-s + (5.63 + 5.63i)9-s + (0.234 + 0.396i)10-s + (−3.50 − 0.275i)11-s + (−6.38 − 1.74i)12-s + (−0.883 − 3.67i)13-s + (3.30 − 2.32i)14-s + (1.07 − 0.0845i)15-s + (3.87 − 0.998i)16-s + (1.39 − 1.19i)17-s + ⋯ |
| L(s) = 1 | + (0.0630 + 0.998i)2-s + (1.76 + 0.731i)3-s + (−0.992 + 0.125i)4-s + (0.129 − 0.0660i)5-s + (−0.618 + 1.80i)6-s + (−0.563 − 0.919i)7-s + (−0.188 − 0.982i)8-s + (1.87 + 1.87i)9-s + (0.0741 + 0.125i)10-s + (−1.05 − 0.0831i)11-s + (−1.84 − 0.503i)12-s + (−0.244 − 1.02i)13-s + (0.882 − 0.620i)14-s + (0.277 − 0.0218i)15-s + (0.968 − 0.249i)16-s + (0.339 − 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00408 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00408 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17391 + 1.17872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17391 + 1.17872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0891 - 1.41i)T \) |
| 41 | \( 1 + (-4.73 + 4.31i)T \) |
| good | 3 | \( 1 + (-3.05 - 1.26i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.290 + 0.147i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (1.49 + 2.43i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (3.50 + 0.275i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.883 + 3.67i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-1.39 + 1.19i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.69 + 0.406i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (3.85 - 2.80i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.55 - 5.59i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (1.45 + 4.48i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.02 - 3.15i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.501 - 0.0794i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.324 + 0.529i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-4.94 + 5.78i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-3.00 - 4.13i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.64 - 0.735i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.0526 + 0.668i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.641 - 8.15i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (9.60 - 9.60i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.664 - 1.60i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 + (-1.48 + 0.908i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.594 + 7.55i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.39865049734502710419940813449, −12.88803492063616311137825104756, −10.29650048032323400452886967105, −9.982814922581610136318694799828, −8.826832883475531087941199825086, −7.88052775558218371928661172981, −7.27400846376964013917902268986, −5.36048087622431250716542992434, −4.06715404791772912507334392330, −3.03295167609317248536397869928,
2.12136388708911411218229808813, 2.81426689300916590832695236004, 4.23631762996635952448861148540, 6.24203767118498248223082478976, 7.84466049668501021186438435272, 8.628892257735746334936567270377, 9.479633710849758779813231217036, 10.28921777243460871591128793931, 12.15148080605963595751302287844, 12.51768107065511500782269173752
