L(s) = 1 | + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4804795171 - 0.3146632793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4804795171 - 0.3146632793i\) |
\(L(1)\) |
\(\approx\) |
\(0.4338633176 - 0.6597335641i\) |
\(L(1)\) |
\(\approx\) |
\(0.4338633176 - 0.6597335641i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.0581 - 0.998i)T \) |
| 5 | \( 1 + (-0.802 + 0.597i)T \) |
| 7 | \( 1 + (0.396 - 0.918i)T \) |
| 11 | \( 1 + (-0.448 + 0.893i)T \) |
| 13 | \( 1 + (0.396 - 0.918i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.998 - 0.0581i)T \) |
| 31 | \( 1 + (0.116 - 0.993i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.727 - 0.686i)T \) |
| 53 | \( 1 + (-0.116 - 0.993i)T \) |
| 59 | \( 1 + (-0.835 + 0.549i)T \) |
| 61 | \( 1 + (0.957 - 0.286i)T \) |
| 67 | \( 1 + (0.802 + 0.597i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.893 - 0.448i)T \) |
| 79 | \( 1 + (0.396 + 0.918i)T \) |
| 83 | \( 1 + (0.0581 - 0.998i)T \) |
| 89 | \( 1 + (-0.957 + 0.286i)T \) |
| 97 | \( 1 + (-0.802 + 0.597i)T \) |
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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
−19.01926193321388003442818263507, −18.17758310095311388107470645242, −17.45562529115539544784184019166, −16.71676247573195600993568984827, −16.05332237414291994289277824594, −15.69156430936990616675567189352, −15.265874007931044766865024860374, −14.428527108438952357538036139, −13.79032518456878180111357074026, −13.027517226795983489399125322, −12.18016321743108975516641163535, −11.28932144111343181758568802042, −11.080141818567604163030860719371, −9.613753415913106215912527926671, −9.12380767549878823872652793613, −8.57798652366362511825009160009, −8.08998165377729772062113526294, −7.22779249402054506956196959762, −6.08825605366177145080827706412, −5.5790490790550403064179256527, −4.859809162954704502972083422982, −4.28930991922470349053107964269, −3.53494641268319801598103873693, −2.80034681342950641677452820482, −1.305071566329491278941675393101,
0.196375239833257784579750673836, 0.87652650706569144622741866570, 2.02140775545851558669560415717, 2.54645347232828828637698686183, 3.51056434072972259362777890223, 4.0710743220734746729852970672, 5.00513239945238653294678585198, 5.81283134769770415252489295066, 6.88050589459471885197627973347, 7.52271508129527778535762377562, 7.99964804103502065082797013492, 8.78828072576057311039753983572, 9.859316457172368654187145610698, 10.58893747800139627785713693704, 11.06557974153455967489814328919, 11.7813563864011730768847165071, 12.362848362079861228145493423, 13.08737398856825132959507781495, 13.636560413968864207419428344657, 14.33988891502509950176368720198, 14.91827191946473335657616943355, 15.697839704967882939128705114493, 16.82918514240328814743210272390, 17.6318960439041241740475541208, 18.13317138128076779515369332545