L(s) = 1 | − 4·5-s + 12·9-s + 24·25-s + 184·29-s − 256·41-s − 48·45-s − 208·49-s − 304·61-s + 90·81-s + 560·89-s − 296·101-s + 608·109-s + 272·121-s − 204·125-s + 127-s + 131-s + 137-s + 139-s − 736·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.18e3·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 4/5·5-s + 4/3·9-s + 0.959·25-s + 6.34·29-s − 6.24·41-s − 1.06·45-s − 4.24·49-s − 4.98·61-s + 10/9·81-s + 6.29·89-s − 2.93·101-s + 5.57·109-s + 2.24·121-s − 1.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.07·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.00·169-s + 0.00578·173-s + 0.00558·179-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) |
≈ |
0.4226111274 |
L(21) |
≈ |
0.4226111274 |
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+2T−6T2+2p2T3+p4T4)2 |
good | 7 | (1+104T2+5454T4+104p4T6+p8T8)2 |
| 11 | (1−136T2+28206T4−136p4T6+p8T8)2 |
| 13 | (1−592T2+144510T4−592p4T6+p8T8)2 |
| 17 | (1−112T2+118878T4−112p4T6+p8T8)2 |
| 19 | (1−436T2+275334T4−436p4T6+p8T8)2 |
| 23 | (1+76pT2+1290726T4+76p5T6+p8T8)2 |
| 29 | (1−46T+1698T2−46p2T3+p4T4)4 |
| 31 | (1−1540T2+1506054T4−1540p4T6+p8T8)2 |
| 37 | (1−4720T2+9299454T4−4720p4T6+p8T8)2 |
| 41 | (1+64T+2334T2+64p2T3+p4T4)4 |
| 43 | (1+4868T2+12528486T4+4868p4T6+p8T8)2 |
| 47 | (1+5540T2+15331014T4+5540p4T6+p8T8)2 |
| 53 | (1−10480T2+43220094T4−10480p4T6+p8T8)2 |
| 59 | (1+2744T2+25695534T4+2744p4T6+p8T8)2 |
| 61 | (1+38T+p2T2)8 |
| 67 | (1+8564T2+44158854T4+8564p4T6+p8T8)2 |
| 71 | (1−3028T2−19410330T4−3028p4T6+p8T8)2 |
| 73 | (1−17140T2+129420582T4−17140p4T6+p8T8)2 |
| 79 | (1−22660T2+205335174T4−22660p4T6+p8T8)2 |
| 83 | (1+23492T2+230783910T4+23492p4T6+p8T8)2 |
| 89 | (1−140T+19830T2−140p2T3+p4T4)4 |
| 97 | (1+8252T2+163205766T4+8252p4T6+p8T8)2 |
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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
−4.04061733237711833512104368735, −4.00085748818378785444668807247, −3.99263539833658227662398411719, −3.50128552138462817245342063226, −3.42380788583507495960345614882, −3.35117014889612471114989863383, −3.19203358740664302770610764924, −3.09347233109709529023541219935, −3.07373298256087246620365470848, −2.96625263476086775497748392040, −2.96198889266008129213025067785, −2.71739902215957690524637652775, −2.30548607902819480527521022445, −2.10787977866707114990332604731, −1.91940676045167751169790719887, −1.91857364169984942664522626921, −1.69725631328386866928097046140, −1.68654242179898356822947790841, −1.41769419244204653472630830900, −1.15757574447309014126204355160, −0.991452294093540837484813135699, −0.75222436773444034239930372398, −0.69440435360764190619165303296, −0.38814877374767682492841676367, −0.04857835803796854480342185229,
0.04857835803796854480342185229, 0.38814877374767682492841676367, 0.69440435360764190619165303296, 0.75222436773444034239930372398, 0.991452294093540837484813135699, 1.15757574447309014126204355160, 1.41769419244204653472630830900, 1.68654242179898356822947790841, 1.69725631328386866928097046140, 1.91857364169984942664522626921, 1.91940676045167751169790719887, 2.10787977866707114990332604731, 2.30548607902819480527521022445, 2.71739902215957690524637652775, 2.96198889266008129213025067785, 2.96625263476086775497748392040, 3.07373298256087246620365470848, 3.09347233109709529023541219935, 3.19203358740664302770610764924, 3.35117014889612471114989863383, 3.42380788583507495960345614882, 3.50128552138462817245342063226, 3.99263539833658227662398411719, 4.00085748818378785444668807247, 4.04061733237711833512104368735
Plot not available for L-functions of degree greater than 10.