L(s) = 1 | + (−0.631 − 2.35i)2-s + (−0.102 + 1.72i)3-s + (−3.42 + 1.97i)4-s + (1.90 − 1.16i)5-s + (4.13 − 0.849i)6-s + (3.36 + 3.36i)8-s + (−2.97 − 0.355i)9-s + (−3.94 − 3.76i)10-s + (3.08 − 1.77i)11-s + (−3.06 − 6.11i)12-s + (−1.28 + 1.28i)13-s + (1.81 + 3.42i)15-s + (1.85 − 3.21i)16-s + (2.95 + 0.792i)17-s + (1.04 + 7.24i)18-s + (0.331 + 0.191i)19-s + ⋯ |
L(s) = 1 | + (−0.446 − 1.66i)2-s + (−0.0593 + 0.998i)3-s + (−1.71 + 0.987i)4-s + (0.853 − 0.520i)5-s + (1.68 − 0.346i)6-s + (1.19 + 1.19i)8-s + (−0.992 − 0.118i)9-s + (−1.24 − 1.18i)10-s + (0.928 − 0.536i)11-s + (−0.884 − 1.76i)12-s + (−0.356 + 0.356i)13-s + (0.469 + 0.883i)15-s + (0.463 − 0.803i)16-s + (0.717 + 0.192i)17-s + (0.245 + 1.70i)18-s + (0.0761 + 0.0439i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.824026 - 0.940701i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.824026 - 0.940701i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.102 - 1.72i)T \) |
| 5 | \( 1 + (-1.90 + 1.16i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.631 + 2.35i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-3.08 + 1.77i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 - 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.95 - 0.792i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.331 - 0.191i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.45 + 0.658i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + (0.323 + 0.561i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.00 + 1.34i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (0.335 - 0.335i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.751 - 2.80i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.815 + 3.04i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.81 + 6.60i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.45 + 9.45i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.31 - 12.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (3.17 + 0.849i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.21 - 1.85i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.51 - 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \) |
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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
−10.17997472336694780257108228243, −9.440600053935769781833044738148, −9.025655316795135530693278496650, −8.222983714231651018506063700127, −6.39317067132851713535959517135, −5.27899356253626243151012826904, −4.32552377426778397583128212750, −3.42590632927523211757433088826, −2.33271358710845863882733742110, −0.948701691874678572072086597427,
1.22362404389451984930923245462, 2.85105769422334782534086908663, 4.79506452388404342544611747271, 5.76887887751908282576034844221, 6.40022925637119947998355346807, 7.08555143368786000566147554472, 7.70800711081887496239531590348, 8.683481406882082974435489287259, 9.470780041992157325072617247593, 10.20382477640616518125311896022