L(s) = 1 | + (−0.449 − 0.893i)2-s + (−0.999 + 0.0372i)3-s + (−0.596 + 0.802i)4-s + (0.980 − 0.197i)5-s + (0.481 + 0.876i)6-s + (0.879 − 0.476i)7-s + (0.984 + 0.172i)8-s + (0.997 − 0.0744i)9-s + (−0.616 − 0.787i)10-s + (0.323 − 0.946i)11-s + (0.566 − 0.824i)12-s + (0.998 − 0.0496i)13-s + (−0.820 − 0.571i)14-s + (−0.972 + 0.233i)15-s + (−0.287 − 0.957i)16-s + (0.867 + 0.498i)17-s + ⋯ |
L(s) = 1 | + (−0.449 − 0.893i)2-s + (−0.999 + 0.0372i)3-s + (−0.596 + 0.802i)4-s + (0.980 − 0.197i)5-s + (0.481 + 0.876i)6-s + (0.879 − 0.476i)7-s + (0.984 + 0.172i)8-s + (0.997 − 0.0744i)9-s + (−0.616 − 0.787i)10-s + (0.323 − 0.946i)11-s + (0.566 − 0.824i)12-s + (0.998 − 0.0496i)13-s + (−0.820 − 0.571i)14-s + (−0.972 + 0.233i)15-s + (−0.287 − 0.957i)16-s + (0.867 + 0.498i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9579482213 - 0.5946525874i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9579482213 - 0.5946525874i\) |
\(L(1)\) |
\(\approx\) |
\(0.8163299726 - 0.3657855033i\) |
\(L(1)\) |
\(\approx\) |
\(0.8163299726 - 0.3657855033i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.449 - 0.893i)T \) |
| 3 | \( 1 + (-0.999 + 0.0372i)T \) |
| 5 | \( 1 + (0.980 - 0.197i)T \) |
| 7 | \( 1 + (0.879 - 0.476i)T \) |
| 11 | \( 1 + (0.323 - 0.946i)T \) |
| 13 | \( 1 + (0.998 - 0.0496i)T \) |
| 17 | \( 1 + (0.867 + 0.498i)T \) |
| 19 | \( 1 + (0.179 + 0.983i)T \) |
| 29 | \( 1 + (-0.0434 + 0.999i)T \) |
| 31 | \( 1 + (-0.834 - 0.551i)T \) |
| 37 | \( 1 + (0.275 + 0.961i)T \) |
| 41 | \( 1 + (-0.635 + 0.771i)T \) |
| 43 | \( 1 + (0.606 + 0.794i)T \) |
| 47 | \( 1 + (0.962 + 0.269i)T \) |
| 53 | \( 1 + (-0.616 + 0.787i)T \) |
| 59 | \( 1 + (-0.381 - 0.924i)T \) |
| 61 | \( 1 + (-0.492 + 0.870i)T \) |
| 67 | \( 1 + (0.783 - 0.621i)T \) |
| 71 | \( 1 + (-0.966 + 0.257i)T \) |
| 73 | \( 1 + (-0.0434 - 0.999i)T \) |
| 79 | \( 1 + (-0.806 - 0.591i)T \) |
| 83 | \( 1 + (-0.885 + 0.465i)T \) |
| 89 | \( 1 + (0.299 - 0.954i)T \) |
| 97 | \( 1 + (0.392 + 0.919i)T \) |
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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
−23.67199615703514445427365631039, −22.946680057289100278515389018373, −22.14729366873306056509710850913, −21.324904151777069580436918306965, −20.38821769481187298227880472629, −18.8186345047937494516117347694, −18.29108641684270211354446513136, −17.552557107781603606691074625162, −17.18746879163759137289517714093, −16.06710185300011191702457577186, −15.302823240137523556160206930175, −14.35424900439302499813408616400, −13.5460713912242161307776259268, −12.47221741712005843130824225184, −11.319824662953017166643407146597, −10.52532712213091241834555132303, −9.603395404340841316427443156581, −8.840723295791404784344381198701, −7.48739433037513145613546192616, −6.78805581818176918844842100042, −5.71854469187803922307094444247, −5.291470150329615456411375791620, −4.23640419545270815995053117371, −2.04495753661317915725213099291, −1.12066930491620691404768424639,
1.22652992145630144823510950916, 1.488919890385612792550846196386, 3.312007602217266637715396170256, 4.33993027898077104728307233732, 5.43845668777229988529993036863, 6.21110658012500488941790071991, 7.64400568185182302012131552671, 8.57245411742530604117262132703, 9.628770971464573287860171755969, 10.55969951983555605504834767906, 11.00376608795775371361435596350, 11.92995099835638412697358968461, 12.86481444618258567146099641098, 13.67391551398868463427473839763, 14.498731373981519137265638878032, 16.29972015029522585775410234110, 16.79611391904618183976264628521, 17.44715737481090144492635255344, 18.424182929355346268613167924767, 18.68748870401818662143472772148, 20.24966747981056040796133981371, 20.93506733316639973870737112928, 21.55452879499074671006812129408, 22.18055864996352937623425850820, 23.259576743592315469280095046970