L(s) = 1 | + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (−0.999 − 0.0190i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (0.749 + 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯ |
L(s) = 1 | + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (−0.999 − 0.0190i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (0.749 + 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.177583034 - 2.224573056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.177583034 - 2.224573056i\) |
\(L(1)\) |
\(\approx\) |
\(1.356081181 - 0.7605795321i\) |
\(L(1)\) |
\(\approx\) |
\(1.356081181 - 0.7605795321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.625 - 0.780i)T \) |
| 5 | \( 1 + (0.969 - 0.244i)T \) |
| 7 | \( 1 + (-0.999 - 0.0190i)T \) |
| 13 | \( 1 + (0.749 + 0.662i)T \) |
| 17 | \( 1 + (-0.610 + 0.791i)T \) |
| 19 | \( 1 + (-0.564 + 0.825i)T \) |
| 23 | \( 1 + (0.327 + 0.945i)T \) |
| 29 | \( 1 + (0.851 - 0.524i)T \) |
| 31 | \( 1 + (0.00951 - 0.999i)T \) |
| 37 | \( 1 + (0.993 + 0.113i)T \) |
| 41 | \( 1 + (-0.161 - 0.986i)T \) |
| 43 | \( 1 + (0.928 - 0.371i)T \) |
| 47 | \( 1 + (0.710 - 0.703i)T \) |
| 53 | \( 1 + (-0.0855 - 0.996i)T \) |
| 59 | \( 1 + (0.935 - 0.353i)T \) |
| 61 | \( 1 + (0.988 - 0.151i)T \) |
| 67 | \( 1 + (-0.888 + 0.458i)T \) |
| 71 | \( 1 + (-0.941 + 0.336i)T \) |
| 73 | \( 1 + (-0.921 - 0.389i)T \) |
| 79 | \( 1 + (0.999 - 0.0380i)T \) |
| 83 | \( 1 + (0.290 + 0.956i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.969 - 0.244i)T \) |
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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
−21.76453791542272160857924310404, −20.82674331414088174341818708792, −20.107290386693052636423863633541, −18.9608602994297691942319564019, −17.95214467162285776822124928668, −17.66230264096321921405901974736, −16.504772542379934587303060434266, −16.04803257196498215853240883994, −15.17911154189657035620788465624, −14.35736177433690017715937541848, −13.49820640623166632626449816154, −13.07147249660568861776821979804, −12.358277490784958239496678345971, −11.087010322954513094358071301410, −10.28685513263323547057320881740, −9.1459564589336650497925654138, −8.732268849366719079080836543806, −7.409340456328684931587582901247, −6.51568629328414868368375690307, −6.19307536694041844901090270159, −5.15609832233311619310552055836, −4.28741045022522342302706705542, −2.951808024352388561919689786968, −2.64769761678762064318332050964, −0.79726839782272999991517339181,
0.667413799848636003401647, 1.77811710504395267584556652378, 2.494648730711423782025825542868, 3.683023801300189294123972925054, 4.30123300610211035504405448462, 5.63357697056290358860591321545, 6.09195806070458511808406658385, 6.8505862367844387398163152822, 8.52898732550814273712080019041, 9.2677971781636903617140777832, 9.98794002884967831978612954838, 10.638572202025990168229759048982, 11.61044194329188458688657796271, 12.53599444829595636872673427966, 13.249702181643014512010066570725, 13.61741651920920399973167973393, 14.56865836143634484767443608679, 15.46648906093359719122900743985, 16.32226183215651174056772218169, 17.22818020258775288287381200165, 18.07464270026857141952168024947, 19.09222112232837056312183369324, 19.36061741330826232968421354747, 20.601612868208294004840632750988, 20.948236074581450670722133852105