L(s) = 1 | + (−0.309 − 0.951i)2-s + (−1.97 + 2.71i)3-s + (−0.809 + 0.587i)4-s + (0.505 − 1.55i)5-s + (3.19 + 1.03i)6-s + (2.28 + 3.14i)7-s + (0.809 + 0.587i)8-s + (−2.56 − 7.88i)9-s − 1.63·10-s + (2.17 + 2.50i)11-s − 3.35i·12-s + (0.670 + 2.06i)13-s + (2.28 − 3.14i)14-s + (3.22 + 4.44i)15-s + (0.309 − 0.951i)16-s + (−2.57 − 0.837i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−1.14 + 1.56i)3-s + (−0.404 + 0.293i)4-s + (0.226 − 0.695i)5-s + (1.30 + 0.423i)6-s + (0.864 + 1.18i)7-s + (0.286 + 0.207i)8-s + (−0.853 − 2.62i)9-s − 0.517·10-s + (0.656 + 0.754i)11-s − 0.969i·12-s + (0.185 + 0.572i)13-s + (0.611 − 0.841i)14-s + (0.833 + 1.14i)15-s + (0.0772 − 0.237i)16-s + (−0.625 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.360544 + 0.582998i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.360544 + 0.582998i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-2.17 - 2.50i)T \) |
| 19 | \( 1 + (3.42 - 2.69i)T \) |
good | 3 | \( 1 + (1.97 - 2.71i)T + (-0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.505 + 1.55i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.28 - 3.14i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.670 - 2.06i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.57 + 0.837i)T + (13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + 7.76T + 23T^{2} \) |
| 29 | \( 1 + (0.839 - 0.609i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (8.28 - 2.69i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.10 - 4.26i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.51 + 1.10i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 4.64iT - 43T^{2} \) |
| 47 | \( 1 + (-2.21 - 1.60i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.32 + 2.70i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.58 + 7.68i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (5.21 + 1.69i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 8.92iT - 67T^{2} \) |
| 71 | \( 1 + (-5.86 - 1.90i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.69 - 9.21i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.147 - 0.454i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.14 - 1.34i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 9.23iT - 89T^{2} \) |
| 97 | \( 1 + (2.85 - 0.928i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.41185367970751868902554022706, −10.72118250231625314789478519036, −9.637389871938144144340684918933, −9.180081427967453509794306639578, −8.407429762705537824497005172618, −6.39111611556094969875540449552, −5.39140792741828406541615650515, −4.65828225150939672795292003097, −3.87700049150286893530545583939, −1.85105434368389385370666994852,
0.55432417750856919830279680710, 1.96683570948206154637823741340, 4.25647065595934782667067374229, 5.65511839681906979479401025010, 6.32001247593638884872998051406, 7.12465636138261773602519372465, 7.73731992798202133648413109035, 8.645477303083648258186964339297, 10.53549153812807499656189725244, 10.84397478931703773639695728903