L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.766 + 0.642i)3-s + (−0.173 − 0.984i)4-s + (−0.342 + 0.939i)5-s − i·6-s + (−0.939 − 0.342i)7-s + (0.866 + 0.5i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)12-s + (0.984 − 0.173i)13-s + (0.866 − 0.5i)14-s + (−0.342 − 0.939i)15-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.766 + 0.642i)3-s + (−0.173 − 0.984i)4-s + (−0.342 + 0.939i)5-s − i·6-s + (−0.939 − 0.342i)7-s + (0.866 + 0.5i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)12-s + (0.984 − 0.173i)13-s + (0.866 − 0.5i)14-s + (−0.342 − 0.939i)15-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1083672477 - 0.08726646715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1083672477 - 0.08726646715i\) |
\(L(1)\) |
\(\approx\) |
\(0.3832365656 + 0.1718731866i\) |
\(L(1)\) |
\(\approx\) |
\(0.3832365656 + 0.1718731866i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.984 - 0.173i)T \) |
| 19 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.939 + 0.342i)T \) |
| 59 | \( 1 + (0.342 + 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.342 + 0.939i)T \) |
| 83 | \( 1 + (0.173 - 0.984i)T \) |
| 89 | \( 1 + (-0.342 - 0.939i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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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
−35.59157530839088270117937176927, −35.084831428031147269693551720102, −33.33352707285324685364141098237, −31.657876353526719133551699401572, −30.53067085196272276371660012087, −29.27821071182184299726585815754, −28.31053915616562671010813054117, −27.79984496189184211004951901853, −25.89138897326668345693189037677, −24.74131724185713264304390485980, −23.153176227630249852833368903952, −22.10424812561150464917726995043, −20.45467546055101580175638251983, −19.38831634440378432213658248056, −18.24592577238899128191072039293, −16.95435636353644813164491998510, −16.02779647559662477994237500788, −13.09939674151563620498986171858, −12.45996389156278132856097994937, −11.24863188243077488114564566937, −9.56497575420961305840147532240, −8.15585646400261687354723655729, −6.41436033326693304967003882318, −4.21308672734253584048682741904, −1.71803990947925726334750393171,
0.12352703136094995655490453035, 3.828297684997818824321245700780, 6.005816163948271956414787896913, 6.86504443277927639152419135198, 8.93800585932748263264698459317, 10.38736269672151304711249796815, 11.27580538890110694348443363863, 13.71468426871066802157244758139, 15.370071905514547620028052535, 16.09411274743672813331450749093, 17.38819153159372671708803747011, 18.61801572486747399031173766379, 19.838781538214457984883660195934, 21.98565570510731059408672399177, 22.90061152868788105013763474816, 23.90886664755232648673333157845, 25.78860146912035121721735002177, 26.57053287049967456267051514182, 27.565861172149412919583947658543, 28.76810500550162904909982751713, 29.94464507181257486098812776037, 32.03850904932065915603768218632, 32.9589002499881259487406769407, 33.99536434134035299953751387736, 34.9878245385283754204293970665