L(s) = 1 | + (−0.999 + 0.0337i)2-s + (−0.184 − 0.982i)3-s + (0.997 − 0.0675i)4-s + (−0.839 − 0.543i)5-s + (0.217 + 0.975i)6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.931 + 0.363i)9-s + (0.857 + 0.514i)10-s + (0.409 + 0.912i)11-s + (−0.250 − 0.968i)12-s + (−0.994 − 0.101i)13-s + (−0.905 + 0.425i)14-s + (−0.378 + 0.925i)15-s + (0.990 − 0.134i)16-s + (−0.440 + 0.897i)17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0337i)2-s + (−0.184 − 0.982i)3-s + (0.997 − 0.0675i)4-s + (−0.839 − 0.543i)5-s + (0.217 + 0.975i)6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.931 + 0.363i)9-s + (0.857 + 0.514i)10-s + (0.409 + 0.912i)11-s + (−0.250 − 0.968i)12-s + (−0.994 − 0.101i)13-s + (−0.905 + 0.425i)14-s + (−0.378 + 0.925i)15-s + (0.990 − 0.134i)16-s + (−0.440 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1552407453 + 0.1395610818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1552407453 + 0.1395610818i\) |
\(L(1)\) |
\(\approx\) |
\(0.4558832202 - 0.1073382866i\) |
\(L(1)\) |
\(\approx\) |
\(0.4558832202 - 0.1073382866i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 373 | \( 1 \) |
good | 2 | \( 1 + (-0.999 + 0.0337i)T \) |
| 3 | \( 1 + (-0.184 - 0.982i)T \) |
| 5 | \( 1 + (-0.839 - 0.543i)T \) |
| 7 | \( 1 + (0.918 - 0.394i)T \) |
| 11 | \( 1 + (0.409 + 0.912i)T \) |
| 13 | \( 1 + (-0.994 - 0.101i)T \) |
| 17 | \( 1 + (-0.440 + 0.897i)T \) |
| 19 | \( 1 + (-0.612 + 0.790i)T \) |
| 23 | \( 1 + (-0.758 - 0.651i)T \) |
| 29 | \( 1 + (-0.985 + 0.168i)T \) |
| 31 | \( 1 + (-0.250 + 0.968i)T \) |
| 37 | \( 1 + (-0.664 - 0.747i)T \) |
| 41 | \( 1 + (-0.954 - 0.299i)T \) |
| 43 | \( 1 + (0.997 - 0.0675i)T \) |
| 47 | \( 1 + (0.0843 + 0.996i)T \) |
| 53 | \( 1 + (-0.931 - 0.363i)T \) |
| 59 | \( 1 + (0.638 + 0.769i)T \) |
| 61 | \( 1 + (-0.184 - 0.982i)T \) |
| 67 | \( 1 + (-0.0506 + 0.998i)T \) |
| 71 | \( 1 + (-0.557 + 0.830i)T \) |
| 73 | \( 1 + (0.780 - 0.625i)T \) |
| 79 | \( 1 + (0.857 + 0.514i)T \) |
| 83 | \( 1 + (-0.931 - 0.363i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.440 + 0.897i)T \) |
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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
−24.33751610536132114137691663866, −23.84872608021832295999181447639, −22.28906094155896536732140124701, −21.826250088762521857532362923701, −20.77215540682530954948143714788, −19.95214476896298233069158339186, −19.153430438703186600902838117396, −18.21630900373038133441800852188, −17.26952205498641412630479206192, −16.53566429571628255899084129114, −15.460306721891807933212259137615, −15.08540274909090261022515428083, −14.070981203659101336022034344585, −12.006642101443167901350079652471, −11.41406812868649598407987970114, −10.93676069681199399541936310342, −9.75348622843986588173796271115, −8.873694777627219651257812631306, −8.03932771224214911501617886622, −7.0121256729143435528718414405, −5.77172031495140251007272740015, −4.54823817252939076868010440081, −3.33200425530100983165427924865, −2.25940218876166088681938217801, −0.17667401818265206841795039691,
1.41215827067488717051119420994, 2.15002834159403306339934248246, 3.99269154969687106113986984820, 5.301940853763550418034530905754, 6.684077303941461616768511372007, 7.50861642124277318958141027037, 8.10893159379390423349265923683, 8.95920827051755732404284197975, 10.39671907043972650517557178878, 11.241992688055082350701696962563, 12.29640376615643198354317599660, 12.54763193652976349330327707137, 14.3933730975263241913473824538, 14.97582993410848142065656105691, 16.29241415387458904656471085325, 17.26947913510835890837592082887, 17.52893211479482321506058384810, 18.69459436472368087417162862184, 19.52414160987752840052360583512, 20.10622925377662137773272762534, 20.84076621717942782248183041948, 22.354453291128852132995148496803, 23.52103496649065011064009220077, 24.129049674242143147312633818869, 24.68713539634598505181409967992