| L(s) = 1 | + (−0.173 − 0.984i)2-s + (1.11 + 1.32i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (1.11 − 1.32i)6-s + (0.652 + 0.237i)7-s + (0.5 + 0.866i)8-s + (−0.520 + 2.95i)9-s + (0.5 − 0.866i)10-s + (1.62 − 1.36i)11-s + (−1.49 − 0.866i)12-s + (−0.532 + 3.01i)13-s + (0.120 − 0.684i)14-s + 1.73i·15-s + (0.766 − 0.642i)16-s + (0.826 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.642 + 0.766i)3-s + (−0.469 + 0.171i)4-s + (0.342 + 0.287i)5-s + (0.454 − 0.541i)6-s + (0.246 + 0.0897i)7-s + (0.176 + 0.306i)8-s + (−0.173 + 0.984i)9-s + (0.158 − 0.273i)10-s + (0.489 − 0.410i)11-s + (−0.433 − 0.249i)12-s + (−0.147 + 0.836i)13-s + (0.0322 − 0.182i)14-s + 0.447i·15-s + (0.191 − 0.160i)16-s + (0.200 − 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48561 + 0.173643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48561 + 0.173643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 7 | \( 1 + (-0.652 - 0.237i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.62 + 1.36i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.532 - 3.01i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.826 + 1.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.29 - 2.24i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.18 + 1.52i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.49 + 8.45i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.06 - 2.20i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.12 + 6.36i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.450 - 0.378i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.75 + 1.73i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + (3.98 + 3.34i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (10.8 + 3.93i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.774 - 4.39i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (5.36 - 9.30i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.86 - 6.69i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0837 + 0.475i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.45 - 8.26i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (6.56 + 11.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 10.1i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.64490988789608411117164272631, −11.00970446222121744802497820541, −9.962466408781774087529006805449, −9.287797109024783468792359933405, −8.470322369710060792398018905120, −7.24483173864364016964988217197, −5.64698137664089888496263871125, −4.40200372634762482763585610349, −3.33027061387772200000394541528, −2.02600829523625448412132962464,
1.40202267496782222944332189169, 3.23016217992112470000019674400, 4.86581299157743409332985341204, 6.05576889320444128321332552137, 7.13157446775848738263547794286, 7.86547872363994154166795798962, 8.922183599886993345332464540376, 9.559018872723210795651183712645, 10.88870956749795405660416151538, 12.22281388126373103034615810529
