L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1571470463 - 2.072425370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1571470463 - 2.072425370i\) |
\(L(1)\) |
\(\approx\) |
\(1.089794606 - 1.191603038i\) |
\(L(1)\) |
\(\approx\) |
\(1.089794606 - 1.191603038i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + (0.766 - 0.642i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
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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
−22.06967571648152879830200890439, −21.42933805276810254170684394789, −20.56962767714864743629279003334, −19.95209017163685513314010695984, −19.00050653710159837552470008867, −17.93063730872812796361106752827, −16.72223824536871494212511269588, −16.39525408520922082064859020075, −15.48290204626681710355350083490, −15.04198958475151802074949152087, −14.17434049504172483220748612369, −13.45421019236148812009594993896, −12.47176092639575339995249655789, −11.630490324315427845765370842165, −11.05094700248280299702432439601, −9.86645817152669586701791197227, −9.064043478984891406173438304376, −8.319332046049521578295850170380, −7.127594812948090409529794975015, −6.15166523507626403455506230993, −5.52996678270203670710826580106, −4.43406007418464830652102272490, −3.986620893279031600327715725417, −2.71798471873622169564117620187, −2.240830076777773555269249617037,
0.56137647186598417617989308865, 1.68783447664415516268436588064, 2.67018603713886218138620302911, 3.513326136751329417557820536990, 4.41511539408639290044550901728, 5.69173304571075812620275055419, 6.26696054903013197988735295062, 7.248575520265545296929177252916, 7.76326937921541350436379523320, 9.086273291648155187153550423445, 10.18953266500258508900094216207, 10.90199427413757143757666033563, 11.80035714417900037267577134960, 12.643779229380669778441687811796, 13.23803025827910490821131984835, 13.7398618790128693520882418633, 14.61716804181424330068129762615, 15.42978300711945541617088643156, 16.33235907059642608376601972169, 17.35923719474229351098219971275, 18.02542222536796421022588672039, 19.20044418010412363879994239029, 19.686352174550626025119444537734, 20.23795131038007909751147342332, 20.993247761266794425072782836422