L(s) = 1 | + (0.965 − 0.258i)2-s + (1.61 − 0.637i)3-s + (0.866 − 0.499i)4-s + (2.88 − 0.772i)5-s + (1.39 − 1.03i)6-s + (−2.53 + 0.754i)7-s + (0.707 − 0.707i)8-s + (2.18 − 2.05i)9-s + (2.58 − 1.49i)10-s + (−0.475 − 0.127i)11-s + (1.07 − 1.35i)12-s + (−3.60 + 0.171i)13-s + (−2.25 + 1.38i)14-s + (4.14 − 3.08i)15-s + (0.500 − 0.866i)16-s + (−0.186 − 0.323i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.929 − 0.368i)3-s + (0.433 − 0.249i)4-s + (1.28 − 0.345i)5-s + (0.567 − 0.421i)6-s + (−0.958 + 0.285i)7-s + (0.249 − 0.249i)8-s + (0.729 − 0.684i)9-s + (0.816 − 0.471i)10-s + (−0.143 − 0.0384i)11-s + (0.310 − 0.391i)12-s + (−0.998 + 0.0476i)13-s + (−0.602 + 0.370i)14-s + (1.07 − 0.795i)15-s + (0.125 − 0.216i)16-s + (−0.0452 − 0.0784i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.88395 - 1.15423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.88395 - 1.15423i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-1.61 + 0.637i)T \) |
| 7 | \( 1 + (2.53 - 0.754i)T \) |
| 13 | \( 1 + (3.60 - 0.171i)T \) |
good | 5 | \( 1 + (-2.88 + 0.772i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.475 + 0.127i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.186 + 0.323i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.747 - 2.78i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.98 - 6.90i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.582iT - 29T^{2} \) |
| 31 | \( 1 + (-4.88 - 1.30i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.74 - 0.734i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (6.61 + 6.61i)T + 41iT^{2} \) |
| 43 | \( 1 + 2.66iT - 43T^{2} \) |
| 47 | \( 1 + (-3.42 - 12.7i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.82 + 1.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.22 + 1.66i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.77 + 6.53i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 0.757i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.908 + 0.908i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.884 + 3.30i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.68 + 9.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.25 - 9.25i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.844 + 3.15i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (12.1 - 12.1i)T - 97iT^{2} \) |
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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
−10.40447836011844938400172039941, −9.668233400630978365781539208368, −9.276943258683120084747333571007, −7.969054386372990755145356865333, −6.91470560308267103204087751269, −6.04324297226636295520589097702, −5.16948328113493566062209829834, −3.71027349727587321572163201281, −2.66576088193795951542696348130, −1.72433559894574444331960686137,
2.26204504866555889502338482041, 2.90815899092765889990905086774, 4.19775107556621463261056814086, 5.25050837831236567369679208594, 6.41660816959991167572343972112, 7.05335853269953278480317211815, 8.248729709722367820219969722584, 9.341347115279688496063839886404, 10.09283226321318920332054019155, 10.46178979287915679999910601653