| L(s) = 1 | + (1.07 + 0.915i)2-s + (−0.935 − 0.186i)3-s + (0.323 + 1.97i)4-s + (0.763 − 0.510i)5-s + (−0.837 − 1.05i)6-s + (0.225 − 0.337i)7-s + (−1.45 + 2.42i)8-s + (−1.93 − 0.799i)9-s + (1.28 + 0.149i)10-s + (−0.892 − 4.48i)11-s + (0.0650 − 1.90i)12-s + (−1.21 − 1.21i)13-s + (0.552 − 0.157i)14-s + (−0.809 + 0.335i)15-s + (−3.79 + 1.27i)16-s + (2.80 + 3.02i)17-s + ⋯ |
| L(s) = 1 | + (0.762 + 0.647i)2-s + (−0.540 − 0.107i)3-s + (0.161 + 0.986i)4-s + (0.341 − 0.228i)5-s + (−0.342 − 0.431i)6-s + (0.0853 − 0.127i)7-s + (−0.515 + 0.856i)8-s + (−0.643 − 0.266i)9-s + (0.407 + 0.0472i)10-s + (−0.269 − 1.35i)11-s + (0.0187 − 0.550i)12-s + (−0.337 − 0.337i)13-s + (0.147 − 0.0420i)14-s + (−0.208 + 0.0865i)15-s + (−0.947 + 0.318i)16-s + (0.679 + 0.733i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03684 + 0.420260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03684 + 0.420260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.07 - 0.915i)T \) |
| 17 | \( 1 + (-2.80 - 3.02i)T \) |
| good | 3 | \( 1 + (0.935 + 0.186i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.763 + 0.510i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.225 + 0.337i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.892 + 4.48i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.21 + 1.21i)T + 13iT^{2} \) |
| 19 | \( 1 + (-1.29 - 3.11i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.243i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (3.14 + 4.71i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (1.67 - 8.41i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 6.56i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 0.743i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (2.03 - 4.91i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-9.03 + 9.03i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.36 + 0.977i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (11.1 + 4.62i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.41 + 3.61i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 8.38T + 67T^{2} \) |
| 71 | \( 1 + (13.3 + 2.65i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (5.16 - 3.45i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (1.81 + 9.10i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (-3.70 + 1.53i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.88 + 6.88i)T - 89iT^{2} \) |
| 97 | \( 1 + (-4.91 - 7.35i)T + (-37.1 + 89.6i)T^{2} \) |
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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
−14.76146978699801754587452241560, −13.91807446792801025681828547033, −12.84342181109101329451082238735, −11.84363863461642863626380881477, −10.74868023706115923592508180782, −8.876771961293412933223475018443, −7.69840398942336806533636702151, −6.04090121009085861616932792344, −5.41461184107572320938829218188, −3.40076588557386504019815978454,
2.51703631973420564195303060679, 4.65589810529638477591915639243, 5.71718135909961012645058548602, 7.20076641737298713286748032010, 9.386756800666021380051732922778, 10.36905536775962192322507665795, 11.50663328568640580837948975229, 12.28962696374896238876653336159, 13.53212283772859055003157255010, 14.50280550485413133429851315377
