L(s) = 1 | + (−0.826 − 0.563i)3-s + (−0.733 + 0.680i)5-s + (−0.5 + 0.866i)7-s + (0.365 + 0.930i)9-s + (0.623 − 0.781i)11-s + (−0.955 − 0.294i)13-s + (0.988 − 0.149i)15-s + (−0.733 − 0.680i)17-s + (−0.365 + 0.930i)19-s + (0.900 − 0.433i)21-s + (0.988 + 0.149i)23-s + (0.0747 − 0.997i)25-s + (0.222 − 0.974i)27-s + (0.826 − 0.563i)29-s + (−0.0747 − 0.997i)31-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)3-s + (−0.733 + 0.680i)5-s + (−0.5 + 0.866i)7-s + (0.365 + 0.930i)9-s + (0.623 − 0.781i)11-s + (−0.955 − 0.294i)13-s + (0.988 − 0.149i)15-s + (−0.733 − 0.680i)17-s + (−0.365 + 0.930i)19-s + (0.900 − 0.433i)21-s + (0.988 + 0.149i)23-s + (0.0747 − 0.997i)25-s + (0.222 − 0.974i)27-s + (0.826 − 0.563i)29-s + (−0.0747 − 0.997i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1884485128 - 0.2998045278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1884485128 - 0.2998045278i\) |
\(L(1)\) |
\(\approx\) |
\(0.5682401407 - 0.07423268547i\) |
\(L(1)\) |
\(\approx\) |
\(0.5682401407 - 0.07423268547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 \) |
good | 3 | \( 1 + (-0.826 - 0.563i)T \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.955 - 0.294i)T \) |
| 17 | \( 1 + (-0.733 - 0.680i)T \) |
| 19 | \( 1 + (-0.365 + 0.930i)T \) |
| 23 | \( 1 + (0.988 + 0.149i)T \) |
| 29 | \( 1 + (0.826 - 0.563i)T \) |
| 31 | \( 1 + (-0.0747 - 0.997i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.623 - 0.781i)T \) |
| 53 | \( 1 + (-0.955 + 0.294i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.0747 - 0.997i)T \) |
| 67 | \( 1 + (0.365 - 0.930i)T \) |
| 71 | \( 1 + (-0.988 + 0.149i)T \) |
| 73 | \( 1 + (-0.955 - 0.294i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.826 + 0.563i)T \) |
| 89 | \( 1 + (-0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.623 - 0.781i)T \) |
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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
−25.09747319509542682851470466152, −23.89706190166179119463621090309, −23.51262359476237983705653819837, −22.53836205353627022735867707231, −21.84104349160721793789442817439, −20.68723478417718832546221030010, −19.85524281708104517382510366903, −19.26647623804875522936383146821, −17.531324748529772044656984485874, −17.17617211974448768997700722249, −16.30457785560308606039167722014, −15.451923643575129706869752964139, −14.61494154506185234999396547291, −13.06865294854000095971660742194, −12.395425788534715647867329015, −11.477171134777197796314441764527, −10.53761114092179317645661925680, −9.58960012492434737847897156974, −8.67624180909920566480134217324, −7.09520372143684352802798307858, −6.62065971998699401128320754859, −4.78477677838932134518197693975, −4.559231716382709846931830429433, −3.332313209120141050972014398760, −1.25989252533101392960045189965,
0.26685720815826150348927522119, 2.18239025139232666686622937411, 3.296585099827032883147276850809, 4.72569666728455543657142415059, 5.91702236360116232864998902318, 6.684362687094832341223635148093, 7.605927084525826942250353393362, 8.7342124848339667675327059425, 10.03298694500352257521963376917, 11.1171816353087856277676546837, 11.83553257354663170060140736860, 12.49729221780667301844823490523, 13.64564946294164503006371833816, 14.80036853762864698231565067317, 15.68467187095343117885786378014, 16.56848126748729577273933379143, 17.50738214327755861493866730754, 18.59058408817172718694219010837, 19.06987305702590783023204281795, 19.7877924486280730537833556086, 21.399684425117761518643992388437, 22.34095032121742995337643737852, 22.635148866654019496754340861350, 23.63982674317607251457907242514, 24.71592520677363460890156370726