L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)10-s + (0.222 − 0.974i)11-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 19-s + 20-s + 22-s + (−0.222 − 0.974i)23-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)10-s + (0.222 − 0.974i)11-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 19-s + 20-s + 22-s + (−0.222 − 0.974i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126830421 + 0.3694445332i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126830421 + 0.3694445332i\) |
\(L(1)\) |
\(\approx\) |
\(0.9683714958 + 0.3554103906i\) |
\(L(1)\) |
\(\approx\) |
\(0.9683714958 + 0.3554103906i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 \) |
good | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + (0.900 + 0.433i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.222 - 0.974i)T \) |
| 29 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 + (-0.900 + 0.433i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.900 + 0.433i)T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.900 - 0.433i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + (-0.623 + 0.781i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + (-0.222 - 0.974i)T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 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
−23.93118823512458678534408569359, −23.58750658966043081792045124311, −22.586211381675283187954339209123, −21.89974195649780203815320570485, −20.798037482823498800937628947566, −20.080924763419151234053928027960, −19.46053163843219409900454683289, −18.376023360140895114652541709913, −17.7559416402867516392773730777, −16.62272151080275195909436500691, −15.22813923179707579149482313062, −14.49963111419731969668465382134, −13.86359430530540582459712220462, −12.37949937411977212992103828322, −11.869413717112171281337143569304, −11.0781307109182623541096199296, −10.08064462658231099413471036202, −9.213710744795164820755885861982, −7.77335839040859235495109037238, −7.23339986692586208716466951202, −5.37002521504899229969102822544, −4.49875759696393233832072000098, −3.638449688092887573448199722393, −2.42447794169163036495337611841, −1.1576091691994638770838416251,
0.85749105889228285381487861525, 2.98724641602235194372247123420, 4.14426379919825066750271694185, 5.090572056260930185931508280682, 5.875708145010753470064694817772, 7.31090754385368815205100759777, 8.07882473272690887522415696388, 8.6378306665584831222735457078, 9.870796208772597634858639164216, 11.376912452457428977477032954856, 12.14389848397243344792686585485, 13.00983085968885963875751128504, 14.39378743304914685669373252113, 14.7108175952445778829570057053, 15.87520622707523022205491777011, 16.4841973270366163406502842432, 17.39808606983356900129115321359, 18.38135272589680073373976377356, 19.19070326489440031379909784506, 20.29767614266204805179456495781, 21.39815256042556233752139140153, 22.11180706928027137456541979696, 23.13647350123562085371229366383, 23.94045337324011725276160227070, 24.581278107486445703283840101694