L(s) = 1 | + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (0.898 − 0.438i)17-s + (0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯ |
L(s) = 1 | + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (0.898 − 0.438i)17-s + (0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1087795256 - 0.2273812713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1087795256 - 0.2273812713i\) |
\(L(1)\) |
\(\approx\) |
\(0.4046616146 - 0.3039530989i\) |
\(L(1)\) |
\(\approx\) |
\(0.4046616146 - 0.3039530989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.694 - 0.719i)T \) |
| 3 | \( 1 + (-0.529 - 0.848i)T \) |
| 7 | \( 1 + (-0.743 - 0.669i)T \) |
| 13 | \( 1 + (0.0697 + 0.997i)T \) |
| 17 | \( 1 + (0.898 - 0.438i)T \) |
| 23 | \( 1 + (0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.882 + 0.469i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.848 + 0.529i)T \) |
| 43 | \( 1 + (-0.342 - 0.939i)T \) |
| 47 | \( 1 + (0.139 - 0.990i)T \) |
| 53 | \( 1 + (0.829 - 0.559i)T \) |
| 59 | \( 1 + (0.990 - 0.139i)T \) |
| 61 | \( 1 + (0.961 - 0.275i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.559 + 0.829i)T \) |
| 73 | \( 1 + (-0.927 - 0.374i)T \) |
| 79 | \( 1 + (-0.241 - 0.970i)T \) |
| 83 | \( 1 + (-0.406 + 0.913i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.694 - 0.719i)T \) |
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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
−22.22696403484436892601727981392, −21.295800610389747544582659753484, −20.456280322768017423393014962653, −19.50878452240291579581817334094, −18.87408183312293821464562973128, −17.87110751107651398188727999373, −17.35236528325931212890683530670, −16.46052141578754949792572371988, −15.89052588445230043857237538223, −15.195180470810452722306409732875, −14.68164080475574255731268465005, −13.45709442895423410312069366578, −12.42467125818403637321441149961, −11.52973365640059629830184944746, −10.51006570571055403984605781238, −10.009920753143140171666075686, −9.21477566475518059989790471512, −8.502312635938084525782825207931, −7.44939352673854902672129457019, −6.43173203365399579167787228172, −5.57781978528593422661511093037, −5.265570347567318481797733760156, −3.8292320563338687975840186358, −2.88124266803385768754299996203, −1.27828703302175854086959062375,
0.16575549537594454259357640684, 1.25036857348148340081812309457, 2.20946268668710652511918850198, 3.26738658588931089618568572015, 4.23941038955035362985016238233, 5.46301453174049162418241027576, 6.83238999700760379591442272180, 7.020632402134166833921136239522, 8.1068560955570381189186641054, 8.988359090534425550730732230348, 9.98766291895363472478603893390, 10.60715855380443541726485676, 11.63245067201623692947911356739, 12.05090266417784745820799439500, 13.09742704779967074847219307799, 13.48052568816794539889938086219, 14.51700785703474873235849928099, 16.0871012718096589142885556237, 16.710850909681347975115881943, 17.05717028828107776354177287654, 18.14661782862902013020313930630, 18.92709897867024548047273719941, 19.10304760659195276531955530443, 20.22367633669488564031614762156, 20.74647547017506672370726244472