L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.35 + 0.779i)3-s + (0.499 + 0.866i)4-s + (−0.882 − 0.509i)5-s + 1.55·6-s − 0.999i·8-s + (−0.284 + 0.492i)9-s + (0.509 + 0.882i)10-s + (−0.510 − 3.27i)11-s + (−1.35 − 0.779i)12-s + 0.167·13-s + 1.58·15-s + (−0.5 + 0.866i)16-s + (1.47 + 2.55i)17-s + (0.492 − 0.284i)18-s + (0.155 − 0.269i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.779 + 0.450i)3-s + (0.249 + 0.433i)4-s + (−0.394 − 0.227i)5-s + 0.636·6-s − 0.353i·8-s + (−0.0947 + 0.164i)9-s + (0.161 + 0.279i)10-s + (−0.153 − 0.988i)11-s + (−0.389 − 0.225i)12-s + 0.0463·13-s + 0.410·15-s + (−0.125 + 0.216i)16-s + (0.358 + 0.620i)17-s + (0.116 − 0.0670i)18-s + (0.0357 − 0.0619i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6390889275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6390889275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.510 + 3.27i)T \) |
good | 3 | \( 1 + (1.35 - 0.779i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.882 + 0.509i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 0.167T + 13T^{2} \) |
| 17 | \( 1 + (-1.47 - 2.55i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 0.269i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.237 + 0.411i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.89iT - 29T^{2} \) |
| 31 | \( 1 + (-2.20 + 1.27i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.04 - 5.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-3.28 - 1.89i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.21 + 7.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.40 - 3.12i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.93 - 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + (-4.85 - 8.40i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.06 - 3.50i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + (5.97 + 3.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−10.14598355599211497682297287156, −9.249320109736040787259566394885, −8.237427430941526756225668519488, −7.88404878244249586384355841766, −6.48959667080888251209099350948, −5.76669971191117415518385360272, −4.73970496755812863440185161434, −3.77131912698459013240984035281, −2.59397420152795211239054213956, −0.904286900774247887509463967626,
0.53140600647998185395252741392, 2.00656896824602390526253713071, 3.48314989032694408959816093009, 4.83270330269191814641526971141, 5.70328478825945639231257978908, 6.55853111221834544083259979645, 7.34043322134695179625843899168, 7.81981121975661663873249790242, 9.100621119346911285056521207040, 9.626854691442960638791594701519