L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.406 + 0.913i)3-s + (−0.669 − 0.743i)4-s − 6-s + (0.933 − 0.358i)7-s + (0.951 − 0.309i)8-s + (−0.669 + 0.743i)9-s + (0.933 − 0.358i)11-s + (0.406 − 0.913i)12-s + (0.629 − 0.777i)13-s + (−0.0523 + 0.998i)14-s + (−0.104 + 0.994i)16-s + (−0.156 − 0.987i)17-s + (−0.406 − 0.913i)18-s + (0.0523 + 0.998i)19-s + ⋯ |
L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.406 + 0.913i)3-s + (−0.669 − 0.743i)4-s − 6-s + (0.933 − 0.358i)7-s + (0.951 − 0.309i)8-s + (−0.669 + 0.743i)9-s + (0.933 − 0.358i)11-s + (0.406 − 0.913i)12-s + (0.629 − 0.777i)13-s + (−0.0523 + 0.998i)14-s + (−0.104 + 0.994i)16-s + (−0.156 − 0.987i)17-s + (−0.406 − 0.913i)18-s + (0.0523 + 0.998i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.110202004 + 1.713794232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110202004 + 1.713794232i\) |
\(L(1)\) |
\(\approx\) |
\(0.9314352491 + 0.6926278776i\) |
\(L(1)\) |
\(\approx\) |
\(0.9314352491 + 0.6926278776i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.406 + 0.913i)T \) |
| 7 | \( 1 + (0.933 - 0.358i)T \) |
| 11 | \( 1 + (0.933 - 0.358i)T \) |
| 13 | \( 1 + (0.629 - 0.777i)T \) |
| 17 | \( 1 + (-0.156 - 0.987i)T \) |
| 19 | \( 1 + (0.0523 + 0.998i)T \) |
| 23 | \( 1 + (0.891 + 0.453i)T \) |
| 29 | \( 1 + (0.406 - 0.913i)T \) |
| 31 | \( 1 + (-0.777 + 0.629i)T \) |
| 37 | \( 1 + (-0.544 + 0.838i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.156 + 0.987i)T \) |
| 47 | \( 1 + (0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.994 - 0.104i)T \) |
| 59 | \( 1 + (-0.207 - 0.978i)T \) |
| 61 | \( 1 + (0.951 - 0.309i)T \) |
| 67 | \( 1 + (-0.994 - 0.104i)T \) |
| 71 | \( 1 + (0.358 + 0.933i)T \) |
| 73 | \( 1 + (-0.987 + 0.156i)T \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + (-0.913 - 0.406i)T \) |
| 89 | \( 1 + (0.358 + 0.933i)T \) |
| 97 | \( 1 + (0.104 + 0.994i)T \) |
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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
−17.73481520772317918862939677278, −17.14738466720353215642072640420, −16.55851438354402636496868239194, −15.30949760880656680517883343520, −14.68956554818777372455270318812, −14.07714432815794639830894524807, −13.47755540396536108287345078488, −12.771757572649881985078421330392, −12.13166920384167930681078012359, −11.652185019965873846076335274051, −11.00399315077156891630718247863, −10.40436768297907816985194758519, −9.0726687393761467885733764292, −8.87019321910204040344945707244, −8.554476644447317367733341352504, −7.24557307173227945872198879651, −7.19357092630685065569802430780, −6.06071318978211312929420867185, −5.159926904039623106789862998652, −4.18845156934066763398548411093, −3.71278047734381136132543796493, −2.681644388183285641857203615117, −1.93982434785690927653678330452, −1.556942760464314082177636138989, −0.6943418760043698780465291743,
0.923389627356180538262502488575, 1.5658372384419746029437643518, 2.86348323828932446307474175713, 3.72550861227009699651758171821, 4.3421211871444122148945942882, 5.08845011240512407083690670611, 5.61336654241766687882237600574, 6.450334186106459437358438280601, 7.34209796137067475741525955665, 8.02919653480604690570027080056, 8.49818732695071596498893036657, 9.15663054063587068216303974214, 9.81787238610929465130174816987, 10.45678423736084237212245856442, 11.17739524122727379043502648963, 11.64458716585684569714101046356, 12.96873444538949450244930967281, 13.74511769933964133906111478870, 14.182442090229171286232252001805, 14.72043740230178280636226460140, 15.31835067922531104373543463222, 15.992272693432393938706069829210, 16.53277789121722280801069361883, 17.18377986556478210531930884333, 17.679550301915569378259168263231