L(s) = 1 | + (0.139 − 0.990i)2-s + (0.970 + 0.241i)3-s + (−0.961 − 0.275i)4-s + (0.374 − 0.927i)6-s + (−0.406 − 0.913i)7-s + (−0.406 + 0.913i)8-s + (0.882 + 0.469i)9-s + (−0.866 − 0.5i)12-s + (0.529 + 0.848i)13-s + (−0.961 + 0.275i)14-s + (0.848 + 0.529i)16-s + (0.469 + 0.882i)17-s + (0.587 − 0.809i)18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (−0.615 + 0.788i)24-s + ⋯ |
L(s) = 1 | + (0.139 − 0.990i)2-s + (0.970 + 0.241i)3-s + (−0.961 − 0.275i)4-s + (0.374 − 0.927i)6-s + (−0.406 − 0.913i)7-s + (−0.406 + 0.913i)8-s + (0.882 + 0.469i)9-s + (−0.866 − 0.5i)12-s + (0.529 + 0.848i)13-s + (−0.961 + 0.275i)14-s + (0.848 + 0.529i)16-s + (0.469 + 0.882i)17-s + (0.587 − 0.809i)18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (−0.615 + 0.788i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.933017049 - 0.7310113759i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.933017049 - 0.7310113759i\) |
\(L(1)\) |
\(\approx\) |
\(1.347370323 - 0.5186219082i\) |
\(L(1)\) |
\(\approx\) |
\(1.347370323 - 0.5186219082i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.139 - 0.990i)T \) |
| 3 | \( 1 + (0.970 + 0.241i)T \) |
| 7 | \( 1 + (-0.406 - 0.913i)T \) |
| 13 | \( 1 + (0.529 + 0.848i)T \) |
| 17 | \( 1 + (0.469 + 0.882i)T \) |
| 23 | \( 1 + (0.342 + 0.939i)T \) |
| 29 | \( 1 + (-0.719 + 0.694i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (0.587 - 0.809i)T \) |
| 41 | \( 1 + (0.241 - 0.970i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (-0.898 + 0.438i)T \) |
| 53 | \( 1 + (0.999 - 0.0348i)T \) |
| 59 | \( 1 + (-0.438 + 0.898i)T \) |
| 61 | \( 1 + (0.615 + 0.788i)T \) |
| 67 | \( 1 + (0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.0348 - 0.999i)T \) |
| 73 | \( 1 + (-0.0697 - 0.997i)T \) |
| 79 | \( 1 + (-0.374 - 0.927i)T \) |
| 83 | \( 1 + (0.207 - 0.978i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.139 + 0.990i)T \) |
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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
−21.65602093829479302368671386173, −20.91751013304442040495315338434, −20.09029815026792933402544211190, −18.95580412276831447510938795009, −18.548521061362729067051327983656, −17.90970455724521708900636902605, −16.74113116750041301147499539978, −15.9339052727633180092318683209, −15.26696552598742578627260143728, −14.74096986232663986852592654149, −13.85043113147596055053426191108, −13.033421679813943035154089822019, −12.60683573447531278027211793300, −11.455469378381597165556467027017, −9.7910834086465135545943977352, −9.555926810541793713164316721867, −8.28988666132518565686902893498, −8.16491898143011703372280483168, −6.96845682274615370449094674378, −6.220846005176426207074034625762, −5.322950230077495220074678442693, −4.269564948000631389178484503868, −3.20969254796266811310253967810, −2.54875472188025892980040796206, −0.88547625089807130383150455596,
1.17781008854958424116638902819, 1.97641615062376477440740926167, 3.2055062429767935841026931601, 3.777621962797920468148819748495, 4.44160717145773576375449869710, 5.65146511259446123487463567768, 6.95053043932742762087211126238, 7.861010716600515227725258758, 8.86995218516856056915208334890, 9.40612590117332276275937158770, 10.36634321445518102576819755699, 10.81622308659349645116834666775, 11.95864980761222014701790169404, 12.94432997880101198157150743274, 13.4824958689349690056631389877, 14.185884453719081158913826283957, 14.83708656447188577713769697586, 15.93327167618696218628134134530, 16.78519792504346605560314881966, 17.730787400336333064376388785369, 18.757091768226531059988838173745, 19.38059969283015889722056604819, 19.82223872635580407806160554660, 20.743844714995604191718250391679, 21.22439885718290722205580416908