L(s) = 1 | + (0.951 − 0.309i)2-s + (0.913 − 0.406i)3-s + (0.809 − 0.587i)4-s + (−0.743 + 0.669i)5-s + (0.743 − 0.669i)6-s + (0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.406 + 0.913i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.406 − 0.913i)18-s + (0.994 + 0.104i)19-s + (−0.207 + 0.978i)20-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.913 − 0.406i)3-s + (0.809 − 0.587i)4-s + (−0.743 + 0.669i)5-s + (0.743 − 0.669i)6-s + (0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.406 + 0.913i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.406 − 0.913i)18-s + (0.994 + 0.104i)19-s + (−0.207 + 0.978i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.939368032 - 1.913864343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.939368032 - 1.913864343i\) |
\(L(1)\) |
\(\approx\) |
\(2.151892694 - 0.7854445733i\) |
\(L(1)\) |
\(\approx\) |
\(2.151892694 - 0.7854445733i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.743 + 0.669i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.994 + 0.104i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.913 - 0.406i)T \) |
| 31 | \( 1 + (0.743 + 0.669i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.994 + 0.104i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.406 - 0.913i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.587 + 0.809i)T \) |
| 61 | \( 1 + (0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.743 + 0.669i)T \) |
| 73 | \( 1 + (-0.406 + 0.913i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (0.207 - 0.978i)T \) |
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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
−21.861100616315698992089243411514, −20.86549006508144446358410003075, −20.47576829335538326165330738703, −19.69745746794112241590383407725, −19.08452317486856150001480973194, −17.73715102772952528263342729679, −16.55363260022453267317790693096, −16.17752188003194513863891081782, −15.343268303513799211817940274141, −14.77464613335765815176323179384, −13.93207337178200175319960790258, −13.15236770581676738018149731076, −12.457020785190902403248267941980, −11.61740172994452213150307987417, −10.68779912468943775401361464417, −9.59058396084164303937868918079, −8.61134230050704535058618297683, −7.83648410628986066633597371881, −7.35356842278115304531622847453, −5.95816167386786390753589523300, −5.07823182755431212032555309970, −4.096672587080992754529853139550, −3.70585348483540051460980278782, −2.62425603683367616331909699149, −1.5081750092516738845789014242,
1.05303640375144155765419696716, 2.31514870922204092848372168864, 3.050147982846288402896685094996, 3.74023669369053378642674361931, 4.63302703179921893049373330588, 5.87314391649031520212025248961, 6.87702679674825765816620097441, 7.466554652110772214593235046519, 8.26004914057004562423969731607, 9.6403691473486237552852001521, 10.21890728194204039259549968627, 11.623713118740073452986855965475, 11.76636872277866539715356738107, 12.89643349277221116379375362867, 13.68032797218612216363842506886, 14.34475101482322193419964106836, 14.917890894960685024759981366748, 15.77100688517870030409198195521, 16.32053585420136949442841880535, 18.04170061604930847539911545781, 18.57222380914680673871791185212, 19.43502044184551818614213175146, 19.984102152037250464221585606241, 20.64527155049196329299510895622, 21.4612599717144066955966389788