L(s) = 1 | + (−0.686 + 0.727i)2-s + (−0.0581 − 0.998i)4-s + (−0.993 − 0.116i)5-s + (0.893 − 0.448i)7-s + (0.766 + 0.642i)8-s + (0.766 − 0.642i)10-s + (0.396 − 0.918i)11-s + (0.973 − 0.230i)13-s + (−0.286 + 0.957i)14-s + (−0.993 + 0.116i)16-s + (0.173 + 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.0581 + 0.998i)20-s + (0.396 + 0.918i)22-s + (0.893 + 0.448i)23-s + ⋯ |
L(s) = 1 | + (−0.686 + 0.727i)2-s + (−0.0581 − 0.998i)4-s + (−0.993 − 0.116i)5-s + (0.893 − 0.448i)7-s + (0.766 + 0.642i)8-s + (0.766 − 0.642i)10-s + (0.396 − 0.918i)11-s + (0.973 − 0.230i)13-s + (−0.286 + 0.957i)14-s + (−0.993 + 0.116i)16-s + (0.173 + 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.0581 + 0.998i)20-s + (0.396 + 0.918i)22-s + (0.893 + 0.448i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6657570992 + 0.02583440739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6657570992 + 0.02583440739i\) |
\(L(1)\) |
\(\approx\) |
\(0.7316518116 + 0.08858051385i\) |
\(L(1)\) |
\(\approx\) |
\(0.7316518116 + 0.08858051385i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.686 + 0.727i)T \) |
| 5 | \( 1 + (-0.993 - 0.116i)T \) |
| 7 | \( 1 + (0.893 - 0.448i)T \) |
| 11 | \( 1 + (0.396 - 0.918i)T \) |
| 13 | \( 1 + (0.973 - 0.230i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.893 + 0.448i)T \) |
| 29 | \( 1 + (-0.286 - 0.957i)T \) |
| 31 | \( 1 + (-0.835 + 0.549i)T \) |
| 37 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (-0.686 - 0.727i)T \) |
| 43 | \( 1 + (0.597 + 0.802i)T \) |
| 47 | \( 1 + (-0.835 - 0.549i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.396 + 0.918i)T \) |
| 61 | \( 1 + (-0.0581 + 0.998i)T \) |
| 67 | \( 1 + (-0.286 + 0.957i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.686 + 0.727i)T \) |
| 83 | \( 1 + (-0.686 + 0.727i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.993 + 0.116i)T \) |
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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
−31.05240217386687407325188163341, −30.00304900693114064296270440019, −28.65931383686744381725539560291, −27.6048923857041673191060129046, −27.248152933820678210103718597724, −25.84186376066469855134319016410, −24.78060966096400860816240471069, −23.2787790132367730871405361935, −22.2974934239045674360248486486, −20.79745093555547472737835925237, −20.28811457527556162086159341728, −18.80359864615634513165452372365, −18.248700203778217888341303182074, −16.84991122498506511608278327624, −15.67154612038522290167264497248, −14.3620215311603872268276264511, −12.58194011160752478817783774810, −11.676755514627017334707635604655, −10.828268129697150915937694861599, −9.23111358826437091777313114294, −8.16593145860477571543289318338, −7.09001228378691396133564310085, −4.71767415251699233223905419943, −3.38749118547016160069622309004, −1.60374649004089874975783527944,
1.147169855495278158729360665145, 3.847060916607352497141398823634, 5.35878487645804882569785278096, 6.915767366585370571515089204729, 8.104699579140437505822357144064, 8.84602284422507120297256237636, 10.759906424188829758194215992185, 11.40526983907931459976114928764, 13.41988975101954881748368576901, 14.68205414654886099322585179580, 15.641384720519435877415059283580, 16.71216163733850617278973185780, 17.73928247019524183672981001724, 19.003161444958792890299674516037, 19.80600856417433526358765978461, 21.08865984991571414035380250033, 22.844911624483944643522945780397, 23.846196706497307170941505247769, 24.38892321329929140873132622063, 25.84705363335257416275338787898, 26.88568301635513074492311657147, 27.57640555911783753004297969782, 28.45714564150378403951349136805, 30.02105735442885264076762837340, 31.0409248027540366922801992774