L(s) = 1 | + (0.965 + 0.258i)5-s + (0.866 + 0.5i)7-s + (−0.258 − 0.965i)11-s − 17-s + (0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + (0.707 + 0.707i)35-s + (−0.707 + 0.707i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (0.5 − 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)5-s + (0.866 + 0.5i)7-s + (−0.258 − 0.965i)11-s − 17-s + (0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + (0.707 + 0.707i)35-s + (−0.707 + 0.707i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (0.5 − 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3631955085 + 1.298632204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3631955085 + 1.298632204i\) |
\(L(1)\) |
\(\approx\) |
\(1.194195063 + 0.1834266026i\) |
\(L(1)\) |
\(\approx\) |
\(1.194195063 + 0.1834266026i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.258 - 0.965i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.965 - 0.258i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.258 + 0.965i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.707 + 0.707i)T \) |
| 59 | \( 1 + (-0.965 - 0.258i)T \) |
| 61 | \( 1 + (0.965 - 0.258i)T \) |
| 67 | \( 1 + (-0.258 + 0.965i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.965 + 0.258i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−17.92031070343599080156367866506, −17.5618886406797716688131139403, −17.09537846041016521640940970659, −15.97894747071841625347602888162, −15.58181433266880320333146006987, −14.45451434044484394598656617542, −14.12030643230699752425334285616, −13.43523931106921969441317723628, −12.67272509952252880941222177689, −12.03701851014793679569389614900, −11.10502121424676899185704351357, −10.41685941086563362065104631583, −9.94165170367710835574366006140, −8.94372248346301762138434610761, −8.53748667355076898339021901232, −7.40718847225964518745106389113, −6.955520514316540612132559248158, −6.0621937691531492176099971471, −5.02451939249091708583366575570, −4.80682572663362329515910129626, −3.850365008908926266772299343540, −2.61882318512317481801387849593, −1.97014342034060744452801827517, −1.28217915988491211420815728771, −0.17749092692518887454017777359,
1.151197777811746564083342049939, 1.87334444156268722860824121331, 2.62299539834155991238967230830, 3.437567389215783785373423359778, 4.522958899732135422658911724430, 5.29709218996622224726795909383, 5.91848603566582111811122987434, 6.48221211666281783952381195367, 7.56213859357743373775545424140, 8.27490504186044260003306247125, 8.88567639039427845981020209342, 9.70938163649117196248422560595, 10.36562263674183212246776962161, 11.17167581690330291492923302411, 11.663855328437285321709617138984, 12.525560883756509365058203801414, 13.52026476181133034899910550747, 13.82257698009779560703717049053, 14.49572343618519457604626203244, 15.31608038307665289909701336346, 15.92930130746564028366039547075, 16.815640440393057787882840942875, 17.42127404051151107740017859149, 18.22314569630545880488365478579, 18.39465963437392663814149247884