L(s) = 1 | + (−0.642 + 0.766i)3-s + (0.342 + 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.984 + 0.173i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)3-s + (0.342 + 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.984 + 0.173i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7796665256 + 0.7266748244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7796665256 + 0.7266748244i\) |
\(L(1)\) |
\(\approx\) |
\(0.8170654701 + 0.2717536823i\) |
\(L(1)\) |
\(\approx\) |
\(0.8170654701 + 0.2717536823i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-0.642 + 0.766i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.342 + 0.939i)T \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.342 + 0.939i)T \) |
| 83 | \( 1 + (0.984 - 0.173i)T \) |
| 89 | \( 1 + (0.342 - 0.939i)T \) |
| 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
−20.53407674389527552288072426089, −19.56074410552114543875085776592, −19.006147256036671590869749788434, −18.07162610438401171303657912808, −17.53459729860189764119604624867, −16.90021440701761595732979984362, −16.14212188710015518065414825672, −15.25224365061331644076356154121, −14.15098003376749436408439958562, −13.62081385349492754982388354604, −12.91136245515333084848847596413, −12.05678075272067223219720592429, −11.28011822685152070837507440943, −10.73879369069274255304156881900, −9.77578293533899840884983324030, −8.835218532355195557146071629614, −7.67950257248032838201341913276, −7.16512609885646145273206683204, −6.637299879910382394366740204473, −5.364583607713960408260699600663, −4.77986623248907270641361375361, −3.84219858629885455060781558708, −2.42429375943152008331814990745, −1.66912816843950606443670644860, −0.55651189403082506702026674580,
0.92140223779599162401554456155, 2.339602154948460865072995653863, 3.29864281038136908596570178185, 4.15905206264333882937492846118, 5.220565337628569961126704852264, 5.79988087910084338697817371067, 6.33216172916735621184223258962, 7.85726112552936763605106326006, 8.4764795052110410743438427725, 9.25820773768089782666261712916, 10.291285530579027244095845518913, 10.851465011735674986644412809393, 11.47310779879384156466330195007, 12.58904044927010500073560864329, 12.868623275083661324427978473527, 14.32141240305635383504863540443, 14.984496935192367493892076795840, 15.50743306979064760902831283942, 16.35963347197677065321817363375, 17.003860684325262969060339448452, 17.778675516132530609649274434941, 18.61270629064514977441861542128, 19.08600857012231105456494120508, 20.495713211381891846161456437946, 20.82490388552817795660569248091