L(s) = 1 | + (−0.457 + 0.889i)2-s + (0.391 + 0.920i)3-s + (−0.581 − 0.813i)4-s + (0.833 − 0.551i)5-s + (−0.997 − 0.0729i)6-s + (−0.872 − 0.489i)7-s + (0.989 − 0.145i)8-s + (−0.694 + 0.719i)9-s + (0.109 + 0.994i)10-s + (−0.0365 + 0.999i)11-s + (0.520 − 0.853i)12-s + (−0.997 + 0.0729i)13-s + (0.833 − 0.551i)14-s + (0.833 + 0.551i)15-s + (−0.322 + 0.946i)16-s + (−0.181 + 0.983i)17-s + ⋯ |
L(s) = 1 | + (−0.457 + 0.889i)2-s + (0.391 + 0.920i)3-s + (−0.581 − 0.813i)4-s + (0.833 − 0.551i)5-s + (−0.997 − 0.0729i)6-s + (−0.872 − 0.489i)7-s + (0.989 − 0.145i)8-s + (−0.694 + 0.719i)9-s + (0.109 + 0.994i)10-s + (−0.0365 + 0.999i)11-s + (0.520 − 0.853i)12-s + (−0.997 + 0.0729i)13-s + (0.833 − 0.551i)14-s + (0.833 + 0.551i)15-s + (−0.322 + 0.946i)16-s + (−0.181 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2167706365 + 0.4146668642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2167706365 + 0.4146668642i\) |
\(L(1)\) |
\(\approx\) |
\(0.5839037587 + 0.4870350789i\) |
\(L(1)\) |
\(\approx\) |
\(0.5839037587 + 0.4870350789i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.457 + 0.889i)T \) |
| 3 | \( 1 + (0.391 + 0.920i)T \) |
| 5 | \( 1 + (0.833 - 0.551i)T \) |
| 7 | \( 1 + (-0.872 - 0.489i)T \) |
| 11 | \( 1 + (-0.0365 + 0.999i)T \) |
| 13 | \( 1 + (-0.997 + 0.0729i)T \) |
| 17 | \( 1 + (-0.181 + 0.983i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.520 + 0.853i)T \) |
| 29 | \( 1 + (-0.694 - 0.719i)T \) |
| 31 | \( 1 + (-0.997 - 0.0729i)T \) |
| 37 | \( 1 + (0.639 - 0.768i)T \) |
| 41 | \( 1 + (-0.694 + 0.719i)T \) |
| 47 | \( 1 + (-0.694 - 0.719i)T \) |
| 53 | \( 1 + (0.989 + 0.145i)T \) |
| 59 | \( 1 + (0.957 + 0.288i)T \) |
| 61 | \( 1 + (-0.872 - 0.489i)T \) |
| 67 | \( 1 + (-0.457 + 0.889i)T \) |
| 71 | \( 1 + (-0.694 + 0.719i)T \) |
| 73 | \( 1 + (-0.581 + 0.813i)T \) |
| 79 | \( 1 + (-0.976 + 0.217i)T \) |
| 83 | \( 1 + (-0.872 - 0.489i)T \) |
| 89 | \( 1 + (0.252 - 0.967i)T \) |
| 97 | \( 1 + (-0.997 - 0.0729i)T \) |
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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
−19.493642131720832870086844789505, −18.879937498415825946338840071240, −18.32260638258332391948376951500, −17.9366754040674981366904972195, −16.82948062001635920369624119750, −16.37309924140873982378982647240, −14.963524457603100496511160478298, −14.18756141551340077187237732132, −13.49663182967178850307911334245, −13.010427260940768694533075756010, −12.200782035363811818283337947080, −11.52108889568732688780217203724, −10.66806795908176617196503654161, −9.68365650237282432989978473511, −9.236359172605798013507485712189, −8.55109464740207917148650935832, −7.37704576934925990017530330499, −6.93882750642794116918150979411, −5.877280652967372491669280088971, −5.07446383984441085521137389969, −3.39715695468934147085410526029, −2.93288810376288656801327605762, −2.38041032457082197701635651172, −1.35097972847872093724685527758, −0.177877574373224716510330506837,
1.45462606153291284166288196077, 2.44953915992460961932554983769, 3.759815566267701949531303784894, 4.52559979606470217949119700909, 5.33907974795015156960311133566, 5.92056230331990269339744110785, 7.088387329104968874082147901805, 7.62951902257862221953880187878, 8.78108017282263501117915495126, 9.37340564242483968941732556623, 10.01105432425994327553013131458, 10.18458274153447107665937310983, 11.49584183129751001813598648194, 12.94308968091157394833436027535, 13.22746533873215019263800563468, 14.24378515456725656229885615594, 14.83040169985547172745686634392, 15.51423245078491336821334077381, 16.32012652566691229284999969197, 16.92386604765214244287280650483, 17.317445439942391588065214536588, 18.17557106430423573453961075223, 19.29899613105203574754079006016, 19.95314506277867251307244721081, 20.302736299959546959225591052863