| L(s) = 1 | + (0.317 + 1.37i)2-s + (−0.239 − 1.71i)3-s + (−1.79 + 0.874i)4-s + (−0.621 + 2.62i)5-s + (2.28 − 0.874i)6-s + (0.844 + 0.628i)7-s + (−1.77 − 2.20i)8-s + (−2.88 + 0.821i)9-s + (−3.81 − 0.0244i)10-s + (2.24 + 2.37i)11-s + (1.93 + 2.87i)12-s + (−5.48 + 2.75i)13-s + (−0.598 + 1.36i)14-s + (4.64 + 0.438i)15-s + (2.47 − 3.14i)16-s + (1.42 + 3.92i)17-s + ⋯ |
| L(s) = 1 | + (0.224 + 0.974i)2-s + (−0.138 − 0.990i)3-s + (−0.899 + 0.437i)4-s + (−0.278 + 1.17i)5-s + (0.934 − 0.356i)6-s + (0.319 + 0.237i)7-s + (−0.627 − 0.778i)8-s + (−0.961 + 0.273i)9-s + (−1.20 − 0.00772i)10-s + (0.675 + 0.716i)11-s + (0.557 + 0.830i)12-s + (−1.52 + 0.763i)13-s + (−0.160 + 0.364i)14-s + (1.20 + 0.113i)15-s + (0.617 − 0.786i)16-s + (0.346 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.341112 + 0.908079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.341112 + 0.908079i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.317 - 1.37i)T \) |
| 3 | \( 1 + (0.239 + 1.71i)T \) |
| good | 5 | \( 1 + (0.621 - 2.62i)T + (-4.46 - 2.24i)T^{2} \) |
| 7 | \( 1 + (-0.844 - 0.628i)T + (2.00 + 6.70i)T^{2} \) |
| 11 | \( 1 + (-2.24 - 2.37i)T + (-0.639 + 10.9i)T^{2} \) |
| 13 | \( 1 + (5.48 - 2.75i)T + (7.76 - 10.4i)T^{2} \) |
| 17 | \( 1 + (-1.42 - 3.92i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.527 - 1.44i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.81 + 2.43i)T + (-6.59 + 22.0i)T^{2} \) |
| 29 | \( 1 + (-3.43 - 5.22i)T + (-11.4 + 26.6i)T^{2} \) |
| 31 | \( 1 + (4.59 + 1.98i)T + (21.2 + 22.5i)T^{2} \) |
| 37 | \( 1 + (5.66 + 4.75i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-11.1 - 0.648i)T + (40.7 + 4.75i)T^{2} \) |
| 43 | \( 1 + (-8.02 - 2.40i)T + (35.9 + 23.6i)T^{2} \) |
| 47 | \( 1 + (3.35 + 7.76i)T + (-32.2 + 34.1i)T^{2} \) |
| 53 | \( 1 + (-1.97 - 1.13i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.18 + 7.61i)T + (-3.43 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-4.29 - 0.501i)T + (59.3 + 14.0i)T^{2} \) |
| 67 | \( 1 + (-1.47 + 2.24i)T + (-26.5 - 61.5i)T^{2} \) |
| 71 | \( 1 + (-0.658 + 3.73i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.20 - 12.5i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.32 - 0.0774i)T + (78.4 - 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.687 - 11.7i)T + (-82.4 + 9.63i)T^{2} \) |
| 89 | \( 1 + (4.10 - 0.723i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.14 - 0.508i)T + (86.6 - 43.5i)T^{2} \) |
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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
−12.32509190408589791372836240061, −11.26351246891494391474746545257, −10.02077747033249544906997578291, −8.855419674581097457325379032068, −7.75200214241919164245112169452, −7.06605649179753946689078299555, −6.50077453792493166359911517984, −5.29483563374901083058724173318, −3.89494743524243034553769780165, −2.28101452274115772355938849136,
0.66537466834894294262904144378, 2.85923347165104824919087813290, 4.17149809844775881749221286313, 4.90021279332229288896783864876, 5.70207615292307935468844570404, 7.80387946481880674590914025163, 8.875371299943757828378518299303, 9.482583884607235769528940717508, 10.34280336021947720017712644593, 11.37611371948550902966725339309
