L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9187333956 - 0.9070267570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9187333956 - 0.9070267570i\) |
\(L(1)\) |
\(\approx\) |
\(0.8777002573 - 0.4918428369i\) |
\(L(1)\) |
\(\approx\) |
\(0.8777002573 - 0.4918428369i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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
−35.123624437194518717405555899967, −34.009948114126599040996765199280, −33.22012397115354004058026326876, −32.01302900634730591563553907790, −30.67135872836632377805446413088, −28.84555918052751054607790768823, −28.14328693319921429710718164482, −26.69022678418406615859446697366, −25.42813983454312159511488769523, −24.82657073062813873791475509252, −23.39353679924658706346235203711, −21.983855020374874777914676311237, −20.62287467979597522593597744459, −18.73180914493737153776103351270, −17.9058345873361278368378105983, −16.7785195279511758675863036186, −15.22429626527556528045924680275, −14.25664297917617296707514238540, −12.654768619306332455475972928234, −10.46851333479332851221914890761, −9.29789717866723374375552268742, −7.91492464863991371554807006731, −6.17390079629104829599285408986, −5.04817757865145753193065436620, −1.87881590272478459477225235084,
1.13661584237586133250823858123, 3.03051772381527771939117114118, 5.08025849488051124824528730656, 7.37198339413050143084043857740, 9.004966630550517714944566873989, 10.297341359147868890087193487925, 11.37454049376359253353363419458, 13.213687945727177998139203674956, 14.01350292153561793815861249016, 16.44370229235040486575902443689, 17.552342349116300763199388354048, 18.5460290501461449316082617394, 20.108446217290996805602298071239, 21.09729152994569869694033184777, 22.06917458748376347714034927810, 23.708765044419139864993265953220, 25.359748225330826742543741490070, 26.506431845375788556147305332761, 27.49979100847375867730585268824, 29.04616301919069493448632123447, 29.60241409737823183259506033967, 30.79251807338368513801682988973, 32.21577619981980030714463134938, 33.547928108528962771863399278889, 34.81412946176364821352328738257