L(s) = 1 | + 4·11-s + 40·19-s + 12·29-s − 40·71-s + 76·79-s + 81-s − 68·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 160·209-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 1.20·11-s + 9.17·19-s + 2.22·29-s − 4.74·71-s + 8.55·79-s + 1/9·81-s − 6.51·109-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 11.0·209-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
Λ(s)=(=((216⋅516⋅118)s/2ΓC(s)8L(s)Λ(2−s)
Λ(s)=(=((216⋅516⋅118)s/2ΓC(s+1/2)8L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
22.30758963 |
L(21) |
≈ |
22.30758963 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 11 | (1−2T+2T2−2pT3+p2T4)2 |
good | 3 | 1−T4−95T8−p4T12+p8T16 |
| 7 | 1−41T4−495T8−41p4T12+p8T16 |
| 13 | 1+46T4+24051T8+46p4T12+p8T16 |
| 17 | 1−601T4+236505T8−601p4T12+p8T16 |
| 19 | (1−5T+pT2)8 |
| 23 | 1+287T4+484593T8+287p4T12+p8T16 |
| 29 | (1−3T+13T2−3pT3+p2T4)4 |
| 31 | (1+41T2+p2T4)4 |
| 37 | 1−1106T4+1569075T8−1106p4T12+p8T16 |
| 41 | (1−57T2+p2T4)4 |
| 43 | 1−5084T4+12089766T8−5084p4T12+p8T16 |
| 47 | 1+3086T4+4580211T8+3086p4T12+p8T16 |
| 53 | 1−881T4−10532295T8−881p4T12+p8T16 |
| 59 | (1−162T2+12179T4−162p2T6+p4T8)2 |
| 61 | (1−173T2+14289T4−173p2T6+p4T8)2 |
| 67 | 1+2404T4−5755290T8+2404p4T12+p8T16 |
| 71 | (1+10T+146T2+10pT3+p2T4)4 |
| 73 | 1+16591T4+122038521T8+16591p4T12+p8T16 |
| 79 | (1−19T+243T2−19pT3+p2T4)4 |
| 83 | 1+5471T4+85671921T8+5471p4T12+p8T16 |
| 89 | (1−333T2+43433T4−333p2T6+p4T8)2 |
| 97 | 1+11239T4+183755865T8+11239p4T12+p8T16 |
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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
−4.05214308377566092779163981436, −3.99804224428995700420988395130, −3.86943465862838171803530231221, −3.86909674073718022070017956713, −3.69230020363874268990680757965, −3.41068936078449281408679539794, −3.37729257873306144054955074759, −3.32768522404168964041972445059, −3.18068389678289552262583963584, −3.02763377765856675825723540275, −2.81327652020980560283863816141, −2.77599928629211002028491996623, −2.74849544593921317877954758797, −2.70065055437919857182417953715, −2.26370423736623481153754182356, −2.09129893194757451022057910227, −1.80313847145750658955185891884, −1.72191252754984689881973519716, −1.38414490653723031343618575146, −1.34240246428388860884425685564, −1.18110071039350545922711705183, −1.04493566107983043255658496641, −0.867601483230896735814702437595, −0.68142534489945786295004895379, −0.47772751160072938499074723736,
0.47772751160072938499074723736, 0.68142534489945786295004895379, 0.867601483230896735814702437595, 1.04493566107983043255658496641, 1.18110071039350545922711705183, 1.34240246428388860884425685564, 1.38414490653723031343618575146, 1.72191252754984689881973519716, 1.80313847145750658955185891884, 2.09129893194757451022057910227, 2.26370423736623481153754182356, 2.70065055437919857182417953715, 2.74849544593921317877954758797, 2.77599928629211002028491996623, 2.81327652020980560283863816141, 3.02763377765856675825723540275, 3.18068389678289552262583963584, 3.32768522404168964041972445059, 3.37729257873306144054955074759, 3.41068936078449281408679539794, 3.69230020363874268990680757965, 3.86909674073718022070017956713, 3.86943465862838171803530231221, 3.99804224428995700420988395130, 4.05214308377566092779163981436
Plot not available for L-functions of degree greater than 10.