L(s) = 1 | + (−0.5 + 0.866i)2-s + i·3-s + (−0.5 − 0.866i)4-s + (0.965 + 0.258i)5-s + (−0.866 − 0.5i)6-s + (−0.707 + 0.707i)7-s + 8-s − 9-s + (−0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.866 − 0.5i)12-s + (−0.965 + 0.258i)13-s + (−0.258 − 0.965i)14-s + (−0.258 + 0.965i)15-s + (−0.5 + 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + i·3-s + (−0.5 − 0.866i)4-s + (0.965 + 0.258i)5-s + (−0.866 − 0.5i)6-s + (−0.707 + 0.707i)7-s + 8-s − 9-s + (−0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.866 − 0.5i)12-s + (−0.965 + 0.258i)13-s + (−0.258 − 0.965i)14-s + (−0.258 + 0.965i)15-s + (−0.5 + 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2594402422 + 0.7349879369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2594402422 + 0.7349879369i\) |
\(L(1)\) |
\(\approx\) |
\(0.4334494341 + 0.5940722424i\) |
\(L(1)\) |
\(\approx\) |
\(0.4334494341 + 0.5940722424i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 11 | \( 1 + (-0.258 + 0.965i)T \) |
| 13 | \( 1 + (-0.965 + 0.258i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.965 + 0.258i)T \) |
| 31 | \( 1 + (-0.965 + 0.258i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + (-0.258 + 0.965i)T \) |
| 53 | \( 1 + (0.258 + 0.965i)T \) |
| 59 | \( 1 + (0.258 + 0.965i)T \) |
| 61 | \( 1 + (-0.866 + 0.5i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.866 - 0.5i)T \) |
| 83 | \( 1 + (-0.707 + 0.707i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + iT \) |
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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
−30.07429917263414858181823928673, −29.412217422411499164068371913939, −29.040659346039378069315066512535, −27.57480952985581281350924932454, −26.15406056245616371993540990927, −25.46481416520837352429153905177, −24.1413859799963492478373263734, −22.82081654370722268822088406573, −21.6591674820136160817231667968, −20.50302000049684546799225609903, −19.34538469302273567507448041479, −18.66307645411419366255085334511, −17.11975650441665710096560741728, −16.90627350784234364804906739207, −14.20322995903301048480076381790, −13.14058308306647420132303831694, −12.5700489746572704999513024878, −10.93506862842109719716041335445, −9.81310185734874644440419634760, −8.506394244286208179097566781925, −7.1830048442011326579160217816, −5.62754791557246409233584397155, −3.34509726957167870114414032789, −1.910786436056288658488203107334, −0.43912953109417743555997322296,
2.51769090977976853740957241101, 4.79234295424188704717668496309, 5.79090004079216100266765916618, 7.14876016774667262476767928875, 9.14022007240134411612905643088, 9.58635823632197694080000091433, 10.70459451598777971644092120707, 12.78853299527779567989954787782, 14.440596652151552384693577432348, 15.07166094214465867264008182629, 16.33883711824105375715829790447, 17.21890882198676418472221146540, 18.346606566038186614130718955904, 19.63619254474296334974462040046, 21.117058322806232320264788055383, 22.24922259038020706290301089984, 23.00452820859742878149367972043, 24.78128807562319805377888358063, 25.663612561262739130925541945992, 26.24076086799771556930879241284, 27.54251289499962970356909370232, 28.44955595515819388375947829628, 29.32928899138088679959583580653, 31.40816077528525494946284945068, 32.22071582833692849497438931023