L(s) = 1 | − 4·9-s − 8·17-s + 26·25-s + 4·41-s − 8·49-s + 88·73-s + 10·81-s − 32·89-s + 112·97-s + 84·113-s + 54·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 4/3·9-s − 1.94·17-s + 26/5·25-s + 0.624·41-s − 8/7·49-s + 10.2·73-s + 10/9·81-s − 3.39·89-s + 11.3·97-s + 7.90·113-s + 4.90·121-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 + 2.58·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
Λ(s)=(=((248⋅38⋅178)s/2ΓC(s)8L(s)Λ(2−s)
Λ(s)=(=((248⋅38⋅178)s/2ΓC(s+1/2)8L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
31.28228629 |
L(21) |
≈ |
31.28228629 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | (1+T2)4 |
| 17 | (1+T)8 |
good | 5 | (1−13T2+84T4−13p2T6+p4T8)2 |
| 7 | (1+2T2+p2T4)4 |
| 11 | (1−27T2+416T4−27p2T6+p4T8)2 |
| 13 | (1−T2+264T4−p2T6+p4T8)2 |
| 19 | (1−35T2+624T4−35p2T6+p4T8)2 |
| 23 | (1+41T2+1404T4+41p2T6+p4T8)2 |
| 29 | (1−88T2+3486T4−88p2T6+p4T8)2 |
| 31 | (1+18T2+p2T4)4 |
| 37 | (1−36T2+950T4−36p2T6+p4T8)2 |
| 41 | (1−T+8T2−pT3+p2T4)4 |
| 43 | (1−11T2+1872T4−11p2T6+p4T8)2 |
| 47 | (1+46T2+p2T4)4 |
| 53 | (1−136T2+9054T4−136p2T6+p4T8)2 |
| 59 | (1−152T2+11550T4−152p2T6+p4T8)2 |
| 61 | (1−72T2+5438T4−72p2T6+p4T8)2 |
| 67 | (1−118T2+p2T4)4 |
| 71 | (1+208T2+19710T4+208p2T6+p4T8)2 |
| 73 | (1−22T+234T2−22pT3+p2T4)4 |
| 79 | (1+146T2+p2T4)4 |
| 83 | (1−pT2)8 |
| 89 | (1+8T+62T2+8pT3+p2T4)4 |
| 97 | (1−14T+pT2)8 |
show more | |
show less | |
L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−3.49883433206534173990013645525, −3.43240745386328841798426135708, −3.29177403388823157296411740125, −3.25283216655049654238260049259, −3.09056784760186995010282760586, −3.07753153563405034143714713971, −2.86142848588378504723207494709, −2.81693907621653321645051906960, −2.58014127433635261974505404119, −2.46371359773653112197591420436, −2.30209271969079575325015830170, −2.28972673909391245697602619061, −2.12873781690962140181878015216, −2.07384946309360936791082612643, −1.96399471624732725125204395400, −1.77013772366427185524971262569, −1.62873536493018476074239795153, −1.44227188342826731851423699486, −1.25283295401248274185005186750, −0.831015361557121323917898635041, −0.73290320524399181261240145902, −0.65165272953014486850634379918, −0.60533029816068475036378735123, −0.60242863589734796587190692136, −0.45475141624519563395985447465,
0.45475141624519563395985447465, 0.60242863589734796587190692136, 0.60533029816068475036378735123, 0.65165272953014486850634379918, 0.73290320524399181261240145902, 0.831015361557121323917898635041, 1.25283295401248274185005186750, 1.44227188342826731851423699486, 1.62873536493018476074239795153, 1.77013772366427185524971262569, 1.96399471624732725125204395400, 2.07384946309360936791082612643, 2.12873781690962140181878015216, 2.28972673909391245697602619061, 2.30209271969079575325015830170, 2.46371359773653112197591420436, 2.58014127433635261974505404119, 2.81693907621653321645051906960, 2.86142848588378504723207494709, 3.07753153563405034143714713971, 3.09056784760186995010282760586, 3.25283216655049654238260049259, 3.29177403388823157296411740125, 3.43240745386328841798426135708, 3.49883433206534173990013645525
Plot not available for L-functions of degree greater than 10.