L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + 11-s − 12-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + 11-s − 12-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.329249534 + 0.9692271098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329249534 + 0.9692271098i\) |
\(L(1)\) |
\(\approx\) |
\(1.065051635 + 0.2301865437i\) |
\(L(1)\) |
\(\approx\) |
\(1.065051635 + 0.2301865437i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 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 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 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 | \( 1 + (0.5 + 0.866i)T \) |
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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
−28.19299238482614882306109467980, −27.021333916898197420578135411574, −25.84572838654910777399172782765, −25.160374631759324407497366622889, −24.40266712716337360064161825750, −23.68875735952803843651342560252, −22.46899821791544170212944789095, −20.83441658562819776779532439586, −19.88097151545249967658264517768, −18.86615532946283567473629616303, −17.97487735421931465989770200735, −16.94552159946511993868638316848, −16.15976646640816874153495707497, −14.54450147945611326704416420161, −13.93394613759313923744449493210, −12.87250024068865215900193741553, −11.5671074377014357590019601936, −9.5827627923429056092062173659, −8.990750769898743501869680725005, −7.93229029240371803477048045257, −6.70699674987566614897807685359, −5.800027830096896743436686448397, −4.215072475319959725669115828899, −1.90387985658066828389503582331, −0.77303868854585844965380616323,
1.710654196008211803867083244911, 3.15249739262704417420450908932, 3.90794084810442651853726912698, 5.710841730203433064096579313356, 7.52534585646528414900724009644, 8.768324460469888012177099581355, 9.762419470966670834232007210739, 10.57042647820377551387644397096, 11.45463986023975682014093797761, 13.03272212244248477673973711749, 14.14216908401251116492844104883, 15.05817963789405006505086850599, 16.52437628874014240985319894383, 17.50038428240896364205073100750, 18.57729747343348117251698827680, 19.66665284131121556503826735785, 20.40377261451916304206913649535, 21.71140269753434983645388353552, 21.972226784848010646198356429076, 23.12647023983958620221793957448, 25.30844176272365758729055426836, 25.69610269957192619173102252859, 26.77905184107202827695739682854, 27.55427767302427694951005350374, 28.33327275277416569124719158969