L(s) = 1 | + (0.173 − 0.984i)2-s + (0.973 − 0.230i)3-s + (−0.939 − 0.342i)4-s + (−0.893 − 0.448i)5-s + (−0.0581 − 0.998i)6-s + (−0.835 + 0.549i)7-s + (−0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.993 − 0.116i)12-s + (−0.0581 − 0.998i)13-s + (0.396 + 0.918i)14-s + (−0.973 − 0.230i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.973 − 0.230i)3-s + (−0.939 − 0.342i)4-s + (−0.893 − 0.448i)5-s + (−0.0581 − 0.998i)6-s + (−0.835 + 0.549i)7-s + (−0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.993 − 0.116i)12-s + (−0.0581 − 0.998i)13-s + (0.396 + 0.918i)14-s + (−0.973 − 0.230i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.5168296966 - 1.311347947i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.5168296966 - 1.311347947i\) |
\(L(1)\) |
\(\approx\) |
\(0.6878011025 - 0.8482763265i\) |
\(L(1)\) |
\(\approx\) |
\(0.6878011025 - 0.8482763265i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (-0.893 - 0.448i)T \) |
| 7 | \( 1 + (-0.835 + 0.549i)T \) |
| 11 | \( 1 + (-0.597 - 0.802i)T \) |
| 13 | \( 1 + (-0.0581 - 0.998i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.597 + 0.802i)T \) |
| 31 | \( 1 + (0.686 - 0.727i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.893 + 0.448i)T \) |
| 53 | \( 1 + (0.993 + 0.116i)T \) |
| 59 | \( 1 + (0.973 + 0.230i)T \) |
| 61 | \( 1 + (0.993 + 0.116i)T \) |
| 67 | \( 1 + (-0.597 - 0.802i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.993 + 0.116i)T \) |
| 79 | \( 1 + (0.396 - 0.918i)T \) |
| 83 | \( 1 + (-0.0581 - 0.998i)T \) |
| 89 | \( 1 + (-0.973 + 0.230i)T \) |
| 97 | \( 1 + (0.286 - 0.957i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.08964252073316283803650236430, −18.19492467242561562032694675433, −17.503258773804435349309500853949, −16.35860714358517780551032968187, −16.19843197317836452321997409944, −15.38590345084717759515152414166, −14.94896905732041728448536759316, −14.28821595614722834139660302302, −13.55945187197844637331134403892, −13.031873207291415829770909383332, −12.30495660426962192722510895173, −11.3864180585695352062010204491, −10.22290898662205797574742136743, −9.79469151192077209153861944901, −9.13697867987431558446176870870, −8.15573828676403340147383138874, −7.720950810333790948323880834056, −7.04331472247483028080534858591, −6.620076214047430470954554432441, −5.440875206326556963746759304716, −4.46145208843726429286672149125, −3.96170227859737792068593446232, −3.37771959651590851331603439145, −2.5775322212010376889649976111, −1.237071145851322826804715243461,
0.404160837124159751171219861821, 1.00939462494186222057083486600, 2.37908817791916027552872174458, 2.99241022869337572152874462797, 3.34845557272028664273783483564, 4.2070547309659497445406618895, 5.16598005036729206542664587701, 5.754283362213333034095058512251, 7.038573518332392465006388154169, 7.80120272238676972507344077733, 8.50864818901566037368025102883, 8.97077426408311563335495643578, 9.734387034396145490166076155739, 10.37212198942340507129792983460, 11.310635841303424192023898375801, 12.020287438330015970495419868814, 12.62593586118165125690302869275, 13.15192414972307528142867545827, 13.67729745338927056734414963496, 14.59760896118027763261046537031, 15.21062763462116105847805563057, 15.9749314480975212150906818923, 16.39236235851172900257747267175, 17.75079490447023965403850406204, 18.51593173355724514137511570734