L(s) = 1 | + (0.716 − 0.697i)2-s + (0.0935 − 0.995i)3-s + (0.0275 − 0.999i)4-s + (0.266 − 0.963i)5-s + (−0.627 − 0.778i)6-s + (−0.340 − 0.940i)7-s + (−0.677 − 0.735i)8-s + (−0.982 − 0.186i)9-s + (−0.480 − 0.876i)10-s + (0.999 − 0.0220i)11-s + (−0.992 − 0.120i)12-s + (0.00551 + 0.999i)13-s + (−0.899 − 0.436i)14-s + (−0.934 − 0.355i)15-s + (−0.998 − 0.0550i)16-s + (0.840 − 0.542i)17-s + ⋯ |
L(s) = 1 | + (0.716 − 0.697i)2-s + (0.0935 − 0.995i)3-s + (0.0275 − 0.999i)4-s + (0.266 − 0.963i)5-s + (−0.627 − 0.778i)6-s + (−0.340 − 0.940i)7-s + (−0.677 − 0.735i)8-s + (−0.982 − 0.186i)9-s + (−0.480 − 0.876i)10-s + (0.999 − 0.0220i)11-s + (−0.992 − 0.120i)12-s + (0.00551 + 0.999i)13-s + (−0.899 − 0.436i)14-s + (−0.934 − 0.355i)15-s + (−0.998 − 0.0550i)16-s + (0.840 − 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3899121914 - 2.006478614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3899121914 - 2.006478614i\) |
\(L(1)\) |
\(\approx\) |
\(0.7325193790 - 1.382406294i\) |
\(L(1)\) |
\(\approx\) |
\(0.7325193790 - 1.382406294i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.716 - 0.697i)T \) |
| 3 | \( 1 + (0.0935 - 0.995i)T \) |
| 5 | \( 1 + (0.266 - 0.963i)T \) |
| 7 | \( 1 + (-0.340 - 0.940i)T \) |
| 11 | \( 1 + (0.999 - 0.0220i)T \) |
| 13 | \( 1 + (0.00551 + 0.999i)T \) |
| 17 | \( 1 + (0.840 - 0.542i)T \) |
| 19 | \( 1 + (-0.917 - 0.396i)T \) |
| 23 | \( 1 + (0.490 + 0.871i)T \) |
| 29 | \( 1 + (0.993 - 0.110i)T \) |
| 31 | \( 1 + (0.546 - 0.837i)T \) |
| 37 | \( 1 + (0.952 - 0.303i)T \) |
| 41 | \( 1 + (0.350 + 0.936i)T \) |
| 43 | \( 1 + (-0.126 + 0.991i)T \) |
| 47 | \( 1 + (0.451 + 0.892i)T \) |
| 53 | \( 1 + (-0.989 - 0.142i)T \) |
| 59 | \( 1 + (0.789 - 0.614i)T \) |
| 61 | \( 1 + (-0.556 + 0.831i)T \) |
| 67 | \( 1 + (0.618 - 0.785i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (-0.739 - 0.673i)T \) |
| 79 | \( 1 + (-0.234 - 0.972i)T \) |
| 83 | \( 1 + (0.988 - 0.153i)T \) |
| 89 | \( 1 + (-0.360 + 0.932i)T \) |
| 97 | \( 1 + (-0.609 + 0.792i)T \) |
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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
−23.32971314954058870982462761374, −22.77169596533715716005958442621, −22.05694650941420315317849612157, −21.602502858863153251428588788581, −20.79440177842644926981848644623, −19.62645072923280875281704155231, −18.65238308190534860117320741379, −17.514312095753935903937803206197, −16.90602256968212034425408556509, −15.87143886710561936464366594357, −15.1252848513158498892521872629, −14.66095727542783831514036712940, −13.97195944829409813550978140906, −12.64148925404502051994010809158, −11.92421787527793511642828743880, −10.79175956524422609893964348889, −9.98063198437376646862135736645, −8.83571589267525655767073984081, −8.15595997643874605368619428589, −6.70600470691193045850995155600, −6.03396557110987659206193184175, −5.2612001003836024874221596413, −4.02825494827388984743816661107, −3.203776899976851522226923235162, −2.458317722896763298148026169862,
0.88371253178235062007037293164, 1.49757530522075937946204472109, 2.749082364585297395055710196879, 3.985872116439820058057084746863, 4.770889491987825019701842231086, 6.10478559813451924187159424036, 6.6563786569159895879093636072, 7.85845035941902594645588664165, 9.18578689881468674734212335798, 9.68054057947018509366392710150, 11.159501455677080125252869342833, 11.815198820473820708472174070212, 12.65322162519303439126185219499, 13.36643774173974597409838389750, 13.96720107195471236659238354326, 14.69043442880785881534713034245, 16.2046513468571271086807457276, 16.973517498886624672701015083130, 17.7647706997421493937361838724, 19.178479000826068870020239134914, 19.40203952403297838940809644554, 20.26186633392915398003072672954, 21.01882119639493307661495356083, 21.86881558835466759517820706048, 23.141912481560794905466687589106