L(s) = 1 | + (0.974 − 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.433 + 0.900i)10-s + (0.781 + 0.623i)11-s − i·12-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.974 + 0.222i)15-s + (0.623 − 0.781i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (0.974 − 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.433 + 0.900i)10-s + (0.781 + 0.623i)11-s − i·12-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.974 + 0.222i)15-s + (0.623 − 0.781i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.336482735 - 0.9923830322i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.336482735 - 0.9923830322i\) |
\(L(1)\) |
\(\approx\) |
\(1.863111678 - 0.5674444185i\) |
\(L(1)\) |
\(\approx\) |
\(1.863111678 - 0.5674444185i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 3 | \( 1 + (0.433 - 0.900i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.781 + 0.623i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.433 - 0.900i)T \) |
| 23 | \( 1 + (-0.222 + 0.974i)T \) |
| 31 | \( 1 + (0.974 - 0.222i)T \) |
| 37 | \( 1 + (0.781 - 0.623i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (-0.974 - 0.222i)T \) |
| 47 | \( 1 + (-0.781 - 0.623i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.433 - 0.900i)T \) |
| 67 | \( 1 + (-0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.974 + 0.222i)T \) |
| 79 | \( 1 + (-0.781 + 0.623i)T \) |
| 83 | \( 1 + (-0.900 + 0.433i)T \) |
| 89 | \( 1 + (0.974 - 0.222i)T \) |
| 97 | \( 1 + (0.433 + 0.900i)T \) |
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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
−37.52829101282501412682340320577, −35.79909569593469150047336538347, −34.3737230564112118469584964880, −32.93999228210607712199348630910, −32.13373860933532408111793833039, −31.60940663267064007496181314545, −29.74818009913736294213063516409, −28.50198626223536218476637900861, −26.979572624964354316361891481803, −25.29240827090451953382618167494, −24.75193013124266946574827913743, −22.77741060955489021162623888160, −21.83840967026622279248960708327, −20.63467801899391642935435791516, −19.627794749167200060024001624399, −16.77487183832076024476457739420, −16.05859785475565978352102140741, −14.66857555159666668385681280792, −13.29217724059807264511985439907, −11.9449356961725847847431895138, −9.914179868587829956569145831799, −8.38030389475862952098068207369, −5.98716994728022731816997731832, −4.57621626133495754081804908699, −2.955261698120846657748266819765,
2.1340826376337406449548185788, 3.69799039299330378979350157956, 6.387389263564517957755702031095, 7.11863534705811442957686118495, 9.81363349136584420519654473246, 11.59886860374900127592551732455, 12.972308637590797923695844339365, 14.08824240930443559458423822691, 15.13933593332879386040185508097, 17.26736173269111935368574092297, 19.20336194434541213821538709993, 19.75980147859914222535972092754, 21.666462301609682914745859493224, 22.82998422810975892378699426484, 23.88099632625181502635767661730, 25.35571381026268010888032397003, 26.20135698442904782383539552225, 28.65809470427005415901725935009, 29.83366549478286309071852784850, 30.40092533147660733581688086032, 31.686679141284306475076800817427, 32.92354006283801009079104039537, 34.24374579899933364274784294576, 35.53328136868934734040255487507, 37.04245994873128389789044407383