L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.959 + 0.281i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (−0.654 + 0.755i)6-s + (0.841 − 0.540i)7-s + (−0.142 − 0.989i)8-s + (0.841 − 0.540i)9-s + (−0.959 − 0.281i)10-s + (−0.654 − 0.755i)11-s + (−0.142 + 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.959 + 0.281i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (−0.654 + 0.755i)6-s + (0.841 − 0.540i)7-s + (−0.142 − 0.989i)8-s + (0.841 − 0.540i)9-s + (−0.959 − 0.281i)10-s + (−0.654 − 0.755i)11-s + (−0.142 + 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0773 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0773 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7629228450 - 0.7060476343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7629228450 - 0.7060476343i\) |
\(L(1)\) |
\(\approx\) |
\(1.014451657 - 0.5314353995i\) |
\(L(1)\) |
\(\approx\) |
\(1.014451657 - 0.5314353995i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 11 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.415 + 0.909i)T \) |
| 19 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 + (-0.959 + 0.281i)T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + T \) |
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
−32.33833163737591062593522252971, −30.87234829963741890266540094052, −30.510309224058408242979785677102, −29.25311526913555245994187826716, −27.837526375885817030255972143890, −26.83748379656782448176186366509, −25.3485826004381647498675220990, −24.291298213782162325754770716631, −23.296805352479392520054996941756, −22.57785648916085293816955584731, −21.610679516487719216214610894476, −20.20067253041817431250862625736, −18.2196477212338562952806809812, −17.75578596922988421800079667193, −16.01397506606287851630723905111, −15.31659049331410424779502665921, −14.01897290411507577553850054701, −12.42372118681701609189725225975, −11.72423767098485407139486636482, −10.52960897400053523623386532578, −7.91050711654474788088693609429, −7.149479512134417585814445696, −5.61500334723326465217175588682, −4.63300632032539017988704011150, −2.67161347791208181140911212010,
1.275049033256970260999779443531, 3.89011457254991300141644975972, 4.80160364296527433802522289728, 6.001777977219258589826835355070, 7.82698509941230163179353287809, 9.87996725670698652632119478358, 11.1991584699270425535702805505, 11.83070095645719680893361147141, 13.06825290412227969721940963997, 14.459752102190001943604782453447, 15.90233471831847773383584933933, 16.69092383871735544522684613458, 18.32492090490889519985611064279, 19.67770794130199995180486917847, 20.971463901243918845792339228302, 21.55089826792661287820622925352, 23.03288408943608549136291032237, 23.91600441646009354919197510738, 24.28088847386893432796787769463, 26.689230053463492995372759896604, 27.72151307169010229166293962669, 28.590092536541131530847194899247, 29.49797146921689141211253614807, 30.71220698803436206196306049067, 31.74321786525679383837461461271