L(s) = 1 | + (1.26 + 0.640i)2-s + (−0.881 − 0.471i)3-s + (1.17 + 1.61i)4-s + (−0.276 + 0.336i)5-s + (−0.810 − 1.15i)6-s + (0.338 + 0.226i)7-s + (0.453 + 2.79i)8-s + (0.555 + 0.831i)9-s + (−0.563 + 0.247i)10-s + (5.34 + 1.62i)11-s + (−0.279 − 1.98i)12-s + (−0.687 + 0.563i)13-s + (0.282 + 0.502i)14-s + (0.402 − 0.166i)15-s + (−1.21 + 3.81i)16-s + (−2.42 − 1.00i)17-s + ⋯ |
L(s) = 1 | + (0.891 + 0.452i)2-s + (−0.509 − 0.272i)3-s + (0.589 + 0.807i)4-s + (−0.123 + 0.150i)5-s + (−0.330 − 0.473i)6-s + (0.128 + 0.0855i)7-s + (0.160 + 0.987i)8-s + (0.185 + 0.277i)9-s + (−0.178 + 0.0782i)10-s + (1.61 + 0.488i)11-s + (−0.0805 − 0.571i)12-s + (−0.190 + 0.156i)13-s + (0.0754 + 0.134i)14-s + (0.103 − 0.0430i)15-s + (−0.304 + 0.952i)16-s + (−0.588 − 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71768 + 1.02861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71768 + 1.02861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.26 - 0.640i)T \) |
| 3 | \( 1 + (0.881 + 0.471i)T \) |
good | 5 | \( 1 + (0.276 - 0.336i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-0.338 - 0.226i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-5.34 - 1.62i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (0.687 - 0.563i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (2.42 + 1.00i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.274 - 2.78i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-2.38 - 0.473i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (2.19 + 7.25i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.54 + 6.54i)T - 31iT^{2} \) |
| 37 | \( 1 + (8.61 - 0.848i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.68 + 8.49i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (3.39 - 1.81i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 5.24i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.0205 + 0.0676i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (2.24 + 1.83i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (2.98 - 5.57i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (3.71 - 6.94i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-5.59 + 8.37i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-8.63 + 5.77i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (1.25 + 3.04i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (0.917 + 0.0903i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (6.22 - 1.23i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (9.59 - 9.59i)T - 97iT^{2} \) |
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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
−11.80535506581929034721007432287, −11.00543654857144247144752977629, −9.660689785883838190516042194626, −8.536654191184328817787588521015, −7.33552634347908189459188218084, −6.69555536542559108118245202835, −5.78657049160035168084792881032, −4.61704877775419918456229344541, −3.69367293231081711887280676495, −1.96869811337174984419513361271,
1.27350750702614013575895151030, 3.13488094841638045860195362219, 4.27975799144668538765972430550, 5.06696768377045298196722661284, 6.36143777730162562192067743707, 6.88018937124325631810482526468, 8.625034238772635646626421664175, 9.531282133572055418663463732269, 10.67255027326099615256093728791, 11.22390929343920410955335522145