L(s) = 1 | + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s − i·12-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s − i·12-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.019526541 - 0.8219827192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.019526541 - 0.8219827192i\) |
\(L(1)\) |
\(\approx\) |
\(1.607775250 - 0.5051594150i\) |
\(L(1)\) |
\(\approx\) |
\(1.607775250 - 0.5051594150i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (0.433 - 0.900i)T \) |
| 3 | \( 1 + (0.781 + 0.623i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.974 - 0.222i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.781 + 0.623i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 31 | \( 1 + (0.433 - 0.900i)T \) |
| 37 | \( 1 + (-0.974 + 0.222i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (-0.433 - 0.900i)T \) |
| 47 | \( 1 + (0.974 + 0.222i)T \) |
| 53 | \( 1 + (-0.900 - 0.433i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.781 + 0.623i)T \) |
| 67 | \( 1 + (0.222 + 0.974i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.433 + 0.900i)T \) |
| 79 | \( 1 + (0.974 - 0.222i)T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.433 - 0.900i)T \) |
| 97 | \( 1 + (0.781 - 0.623i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−36.62206442634949235245163788668, −36.14000881168350163576523553163, −34.50000054051843011200847399237, −33.478261346335069698301720203178, −31.98318786162667095218734398665, −31.295303298256065065819407533168, −30.01747036729571956083038243042, −28.408289135711714958968182789596, −26.47143978210190184320581019194, −25.469005199037844790533722915197, −24.57250497752782310610425371287, −23.616881345228886736821371650429, −21.61352561586080626114205134354, −20.76948549474588535844032025366, −18.52788973054692222833966804559, −17.68173701727030892970621120911, −15.89040691336259250312144215255, −14.44829687946804026573966876336, −13.48026980732169470106766841378, −12.25367715325095036270516538500, −9.264223853844620541376941805765, −8.21062628537667045847387786286, −6.53123332195502238356460766533, −4.90641606986875696017892434199, −2.37695957689145852606659833082,
2.105384994079940654353416695618, 3.75069900661620720258363749484, 5.490326561383460650308063631744, 8.25947504168891974121934927231, 10.17015652356203222584299064695, 10.657797171874414295373478706463, 13.12162943458100624006570019854, 14.05246429032412334651953518349, 15.2142241317931708151866077725, 17.48767435345017586633011991631, 18.96065385844321962134594341719, 20.49637552452733423484022592001, 21.14006577684221171685518531273, 22.335333892828165735875678951393, 23.91557932122032153280770622949, 25.68584763147432107595705059387, 26.8497754041585174163435036840, 28.076983326652216398797951535798, 29.69483416569308317921767748891, 30.46878692734011666661714303969, 31.832162750035468301585919084421, 32.90204232055572251251634565273, 33.810627037120124508961472958701, 36.3887787775696410616030558024, 37.12576164201851730220604498195