L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 3.73·5-s + (−0.499 + 0.866i)6-s + (1.36 − 2.36i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.86 − 3.23i)10-s + (−0.633 − 1.09i)11-s + 0.999·12-s − 2.73·14-s + (−1.86 − 3.23i)15-s + (−0.5 − 0.866i)16-s + (2.86 − 4.96i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.66·5-s + (−0.204 + 0.353i)6-s + (0.516 − 0.894i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.590 − 1.02i)10-s + (−0.191 − 0.331i)11-s + 0.288·12-s − 0.730·14-s + (−0.481 − 0.834i)15-s + (−0.125 − 0.216i)16-s + (0.695 − 1.20i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.641871938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.641871938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.86 + 4.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.09 + 3.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + (1.76 + 3.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.69 - 8.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.83 + 8.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 7.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.56 + 11.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.36 - 4.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 0.196T + 83T^{2} \) |
| 89 | \( 1 + (-4.73 - 8.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.894298628793298207221251997172, −9.059894377773972190069464777896, −8.088588779179650751188415835641, −7.24475069030152342471311995898, −6.26613876679924249381235977495, −5.45045609462457575957114398073, −4.46290468013006483651713456094, −2.95819985223216164617315160464, −1.92012898038350777096510378350, −0.950900944682986773641632937442,
1.57571721381686050679496760927, 2.60764351709029808494873694966, 4.37724996921088537345360347029, 5.38068082764858988781042809711, 5.84494241582686315342978736907, 6.53760685400302860304977496968, 7.81546186649155360409167950632, 8.763872962134815614524881739550, 9.332847697381587169536813083680, 10.10258298234398457865634786862