L(s) = 1 | + (0.677 + 0.735i)2-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (−0.789 + 0.614i)8-s + (−0.789 + 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.245 + 0.969i)13-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + (0.789 + 0.614i)19-s + (−0.986 − 0.164i)20-s + (0.879 + 0.475i)22-s + (−0.789 − 0.614i)23-s + (−0.986 − 0.164i)25-s + (−0.879 + 0.475i)26-s + ⋯ |
L(s) = 1 | + (0.677 + 0.735i)2-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (−0.789 + 0.614i)8-s + (−0.789 + 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.245 + 0.969i)13-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + (0.789 + 0.614i)19-s + (−0.986 − 0.164i)20-s + (0.879 + 0.475i)22-s + (−0.789 − 0.614i)23-s + (−0.986 − 0.164i)25-s + (−0.879 + 0.475i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1802205948 + 0.05288060378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1802205948 + 0.05288060378i\) |
\(L(1)\) |
\(\approx\) |
\(0.8730000813 + 0.8858113231i\) |
\(L(1)\) |
\(\approx\) |
\(0.8730000813 + 0.8858113231i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (0.677 + 0.735i)T \) |
| 5 | \( 1 + (-0.0825 + 0.996i)T \) |
| 11 | \( 1 + (0.945 - 0.324i)T \) |
| 13 | \( 1 + (-0.245 + 0.969i)T \) |
| 17 | \( 1 + (-0.401 + 0.915i)T \) |
| 19 | \( 1 + (0.789 + 0.614i)T \) |
| 23 | \( 1 + (-0.789 - 0.614i)T \) |
| 29 | \( 1 + (-0.986 + 0.164i)T \) |
| 31 | \( 1 + (-0.879 + 0.475i)T \) |
| 37 | \( 1 + (0.879 + 0.475i)T \) |
| 41 | \( 1 + (0.677 - 0.735i)T \) |
| 43 | \( 1 + (0.546 - 0.837i)T \) |
| 47 | \( 1 + (-0.945 + 0.324i)T \) |
| 53 | \( 1 + (0.945 - 0.324i)T \) |
| 59 | \( 1 + (-0.879 + 0.475i)T \) |
| 61 | \( 1 + (-0.401 - 0.915i)T \) |
| 67 | \( 1 + (-0.401 - 0.915i)T \) |
| 71 | \( 1 + (-0.677 + 0.735i)T \) |
| 73 | \( 1 + (0.945 + 0.324i)T \) |
| 79 | \( 1 + (-0.986 - 0.164i)T \) |
| 83 | \( 1 + (-0.789 + 0.614i)T \) |
| 89 | \( 1 + (-0.546 - 0.837i)T \) |
| 97 | \( 1 + (0.879 + 0.475i)T \) |
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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
−17.97965942798250838754369089535, −16.987053891902652003566642081219, −16.20628884649774462013739745432, −15.58323747278503264891025482901, −14.85438427631212355857051799124, −14.19662777471658708940473277210, −13.27109750506004811392155642829, −13.03881887510921440584998221347, −12.163148241386115993156562331488, −11.594367095563764568889140197790, −11.116823639696543470118991437410, −9.959518503629217892120472697821, −9.40872812342791432690217075511, −9.03668108971834935095466894523, −7.81653622805790830769268003439, −7.192219197326481679744177790106, −6.00264072755056000705574367417, −5.54702155141682205898321364935, −4.70148740447329945139878749374, −4.17254276400927306328331994117, −3.33502625761451428798154911442, −2.45883352137073970888595680768, −1.55364814338607300661658127112, −0.826915515192429409216782047120, −0.02331582260702222499307324846,
1.645207508269217795853290054141, 2.40848467584523413135502873376, 3.5076265680981214750404372703, 3.854665291263662194388714649093, 4.63504888587434064445608990336, 5.87281951601258940858513194697, 6.12749386411290540377094647139, 7.005107095196649188934028880499, 7.44665941386877569221688092829, 8.36052954344939669964113850352, 9.093777120249726788789565967178, 9.85588963810143467565938452477, 10.90323175475238471251299843040, 11.46770795070523044829240921837, 12.13028072544275824292138028218, 12.81773135216245367959708749041, 13.84247056749501644448509122090, 14.22995620105414526327522051706, 14.70053748020566520784780463288, 15.38848903054082657911386187542, 16.22920988692391808794731322066, 16.74237093463241529774320926651, 17.4029494431994158733289661035, 18.26358829912541133845743173797, 18.69780281307849469031397411208