L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)28-s + (0.5 + 0.866i)29-s − i·31-s + (0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)28-s + (0.5 + 0.866i)29-s − i·31-s + (0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7224752272 + 0.3987257138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7224752272 + 0.3987257138i\) |
\(L(1)\) |
\(\approx\) |
\(0.7540055343 + 0.2329068299i\) |
\(L(1)\) |
\(\approx\) |
\(0.7540055343 + 0.2329068299i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 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
−26.615112416453629269982743361485, −26.46536907875145392018231713367, −25.03933267037368793079441466987, −24.19070413233920145211080566622, −23.10674824216080661042721725524, −21.65618648374748390339854728803, −21.03894304135804381411490574480, −20.15391282649519742021646406816, −19.134641662308088646056679814300, −18.19409029139475771203514522034, −17.39505540265263322954696828403, −16.451917306658588509295487776686, −15.413741462008383498514157865434, −14.05666512534262680157119460045, −12.94113577992954095854790350586, −11.79863644943899044243645832847, −10.80901146830070208585803480338, −10.1384054636680703538197739374, −8.67664405519975396371158359949, −7.953997317310953428936264137254, −6.88534915531915462911463381531, −5.226910155547674524294639965092, −3.75823815407918079168835147559, −2.4409964922988763156805640598, −0.99628479471565851305740786538,
1.397585913064869499700104611140, 2.7484056417810336456903031424, 4.92804310702322946306392730430, 5.676186363351765547399247265495, 7.26640427092044111187235844616, 7.92769557458567971496597598363, 9.10063715511244686891820112535, 10.06733741141620636044831885939, 11.18840200019664910414565512997, 12.121545548764343682794526639300, 13.74135637452607346895435221559, 14.76782868374278147364625885831, 15.6070523536056925987483659567, 16.52610671314099830256299693214, 17.79363087098301786198009119218, 18.226044243496420608740506303539, 19.246178951777605741027479440233, 20.51979338837583386726658863670, 21.074996741589023373772287020764, 22.639206916943807892370352448725, 23.639451173308039319626587187460, 24.494872422536530771184710698423, 25.32374069455042481185552741854, 26.17694411772961992468984917008, 27.28629468634343745255906855396