L(s) = 1 | + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + 9-s + 10-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s − 19-s + (−0.5 − 0.866i)20-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + 9-s + 10-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s − 19-s + (−0.5 − 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3101297164 - 0.2766675687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3101297164 - 0.2766675687i\) |
\(L(1)\) |
\(\approx\) |
\(0.4713682522 - 0.1168838833i\) |
\(L(1)\) |
\(\approx\) |
\(0.4713682522 - 0.1168838833i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.0111299046494014155207060499, −21.172041328353629020151711032850, −19.98912061612869246769819006296, −19.43913167967740077075255833352, −18.3859700992036177930814594918, −17.789711615885091025517772015606, −17.04645205821174520142117924532, −16.315588540295000243598740349542, −15.86828112240136692676337851349, −15.11120085878073114872136200248, −13.9674261873820970298663730769, −13.03051470010562948204091471206, −12.313058051966733984524450901049, −11.305868876603608008400753048532, −10.61461561339160024792036891377, −9.54812653766253565390807485667, −8.88949378342201268927227599148, −7.884220623580766546696717096380, −7.12057949761714994686058730529, −6.277964492260114832636988883702, −5.3231846887392909341999436583, −4.753555929153386743345097237037, −3.87434139269327610395652398906, −1.82520592912738855086932058207, −0.76379218205269310586329319273,
0.36621916874900506272839100332, 1.81054237442315014363630954122, 2.742490257872930938427412940848, 4.08335311746112999501515921631, 4.399494628791969713598054962239, 5.98586988696731931431120550332, 6.69937899667177243452546242716, 7.71076755956890698801879606115, 8.46117384138943318566122501770, 9.7504535362563333501024964634, 10.41315851721412830256372885507, 11.07136565121951179998477977881, 11.60878520101236615701324508916, 12.53758023039475969772660386776, 13.10832719497066617919169731678, 14.329312712927029100167374953061, 15.29735747819321158197103373248, 16.164498656131387252649317661717, 17.04134336658617958333410938901, 17.670349521727603086805674819, 18.40944227035578238427728859651, 19.08085690539646420144734767808, 19.65393061708941344705860722747, 20.80258736509189415087126202112, 21.52409258518110435653204739091