L(s) = 1 | − 12·9-s + 16·13-s − 16·17-s + 20·25-s + 96·29-s + 112·37-s + 112·41-s + 104·49-s − 224·53-s − 80·61-s + 272·73-s + 90·81-s − 48·89-s + 528·97-s + 416·101-s − 112·109-s − 176·113-s − 192·117-s + 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 192·153-s + 157-s + ⋯ |
L(s) = 1 | − 4/3·9-s + 1.23·13-s − 0.941·17-s + 4/5·25-s + 3.31·29-s + 3.02·37-s + 2.73·41-s + 2.12·49-s − 4.22·53-s − 1.31·61-s + 3.72·73-s + 10/9·81-s − 0.539·89-s + 5.44·97-s + 4.11·101-s − 1.02·109-s − 1.55·113-s − 1.64·117-s + 0.859·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.25·153-s + 0.00636·157-s + ⋯ |
Λ(s)=(=((248⋅38⋅58)s/2ΓC(s)8L(s)Λ(3−s)
Λ(s)=(=((248⋅38⋅58)s/2ΓC(s+1)8L(s)Λ(1−s)
Particular Values
L(23) |
≈ |
13.85638325 |
L(21) |
≈ |
13.85638325 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | (1+pT2)4 |
| 5 | (1−pT2)4 |
good | 7 | 1−104T2+3804T4−61144T6+732230T8−61144p4T10+3804p8T12−104p12T14+p16T16 |
| 11 | 1−104T2+25116T4+998312T6+87392006T8+998312p4T10+25116p8T12−104p12T14+p16T16 |
| 13 | (1−8T+396T2−4792T3+76550T4−4792p2T5+396p4T6−8p6T7+p8T8)2 |
| 17 | (1+8T−84T2−3848T3−35290T4−3848p2T5−84p4T6+8p6T7+p8T8)2 |
| 19 | 1−200T2+273628T4−52427000T6+42482298118T8−52427000p4T10+273628p8T12−200p12T14+p16T16 |
| 23 | 1−1544T2+1318044T4−803710264T6+443101191110T8−803710264p4T10+1318044p8T12−1544p12T14+p16T16 |
| 29 | (1−24T+1646T2−24p2T3+p4T4)4 |
| 31 | 1−2824T2+6208540T4−8647788856T6+9849609291334T8−8647788856p4T10+6208540p8T12−2824p12T14+p16T16 |
| 37 | (1−56T+3948T2−157576T3+7918598T4−157576p2T5+3948p4T6−56p6T7+p8T8)2 |
| 41 | (1−56T+2556T2−74632T3+3542726T4−74632p2T5+2556p4T6−56p6T7+p8T8)2 |
| 43 | 1−10760T2+54445788T4−171718842040T6+375491195821958T8−171718842040p4T10+54445788p8T12−10760p12T14+p16T16 |
| 47 | 1−10888T2+61255708T4−225571130296T6+586057028916550T8−225571130296p4T10+61255708p8T12−10888p12T14+p16T16 |
| 53 | (1+112T+13020T2+887312T3+57092774T4+887312p2T5+13020p4T6+112p6T7+p8T8)2 |
| 59 | 1−20456T2+200671260T4−1224887792344T6+1465211341814p2T8−1224887792344p4T10+200671260p8T12−20456p12T14+p16T16 |
| 61 | (1+40T+11964T2+456920T3+61872806T4+456920p2T5+11964p4T6+40p6T7+p8T8)2 |
| 67 | 1−4744T2+570460T4+34360533704T6−13806137860346T8+34360533704p4T10+570460p8T12−4744p12T14+p16T16 |
| 71 | 1−19720T2+218875804T4−1700648520760T6+9880226933129926T8−1700648520760p4T10+218875804p8T12−19720p12T14+p16T16 |
| 73 | (1−136T+19228T2−1717816T3+139076998T4−1717816p2T5+19228p4T6−136p6T7+p8T8)2 |
| 79 | 1−36104T2+626466076T4−6818196448568T6+50837338178201926T8−6818196448568p4T10+626466076p8T12−36104p12T14+p16T16 |
| 83 | 1−29320T2+467847004T4−4970634792760T6+39363161035770886T8−4970634792760p4T10+467847004p8T12−29320p12T14+p16T16 |
| 89 | (1+24T+4060T2−454104T3−10096506T4−454104p2T5+4060p4T6+24p6T7+p8T8)2 |
| 97 | (1−132T+20102T2−132p2T3+p4T4)4 |
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L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.20264690686788907540007041479, −3.88393883557736479365470427748, −3.84871819377706010112035865623, −3.54310821417531300301322100590, −3.45995151985215168926365174377, −3.43457630346128456465185052877, −3.27062668539582058117309358556, −3.15726306005533341460631365865, −2.98299728832251637369083801920, −2.89027555929545871450418741713, −2.58767143801005855265787935310, −2.49104460086305845519706580189, −2.48108494750059151968467693901, −2.22115707467574197607466549765, −2.17471782505371670969584439017, −2.16780680631642085178827553249, −1.70438684797098199428260008406, −1.47685674435094951760060751857, −1.37959989915866487133945443283, −1.04871616666934553544796421107, −0.937341432871829972309011356238, −0.815599122447077493222529804736, −0.68537660877571038723214623289, −0.43065757070828910208124036501, −0.26917009535345646624705867534,
0.26917009535345646624705867534, 0.43065757070828910208124036501, 0.68537660877571038723214623289, 0.815599122447077493222529804736, 0.937341432871829972309011356238, 1.04871616666934553544796421107, 1.37959989915866487133945443283, 1.47685674435094951760060751857, 1.70438684797098199428260008406, 2.16780680631642085178827553249, 2.17471782505371670969584439017, 2.22115707467574197607466549765, 2.48108494750059151968467693901, 2.49104460086305845519706580189, 2.58767143801005855265787935310, 2.89027555929545871450418741713, 2.98299728832251637369083801920, 3.15726306005533341460631365865, 3.27062668539582058117309358556, 3.43457630346128456465185052877, 3.45995151985215168926365174377, 3.54310821417531300301322100590, 3.84871819377706010112035865623, 3.88393883557736479365470427748, 4.20264690686788907540007041479
Plot not available for L-functions of degree greater than 10.