L(s) = 1 | + 6·9-s − 4·19-s − 8·25-s + 12·29-s − 8·31-s + 6·49-s − 12·59-s + 40·61-s + 48·71-s + 8·79-s + 17·81-s + 24·101-s + 4·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s − 24·171-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 2·9-s − 0.917·19-s − 8/5·25-s + 2.22·29-s − 1.43·31-s + 6/7·49-s − 1.56·59-s + 5.12·61-s + 5.69·71-s + 0.900·79-s + 17/9·81-s + 2.38·101-s + 0.383·109-s − 3.63·121-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 + 1.38·169-s − 1.83·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
Λ(s)=(=((216⋅54⋅194)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((216⋅54⋅194)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅54⋅194
|
Sign: |
1
|
Analytic conductor: |
21701.1 |
Root analytic conductor: |
3.48385 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅54⋅194, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
5.001029654 |
L(21) |
≈ |
5.001029654 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C22 | 1+8T2+p2T4 |
| 19 | C1 | (1+T)4 |
good | 3 | D4×C2 | 1−2pT2+19T4−2p3T6+p4T8 |
| 7 | D4×C2 | 1−6T2+5pT4−6p2T6+p4T8 |
| 11 | C22 | (1+20T2+p2T4)2 |
| 13 | D4×C2 | 1−18T2+131T4−18p2T6+p4T8 |
| 17 | C22 | (1−33T2+p2T4)2 |
| 23 | D4×C2 | 1−6T2−733T4−6p2T6+p4T8 |
| 29 | D4 | (1−6T+59T2−6pT3+p2T4)2 |
| 31 | D4 | (1+4T+48T2+4pT3+p2T4)2 |
| 37 | C22 | (1−2T2+p2T4)2 |
| 41 | C22 | (1+64T2+p2T4)2 |
| 43 | D4×C2 | 1−64T2+2130T4−64p2T6+p4T8 |
| 47 | C2 | (1−pT2)4 |
| 53 | D4×C2 | 1−50T2+3651T4−50p2T6+p4T8 |
| 59 | D4 | (1+6T+29T2+6pT3+p2T4)2 |
| 61 | D4 | (1−20T+204T2−20pT3+p2T4)2 |
| 67 | D4×C2 | 1−70T2+4371T4−70p2T6+p4T8 |
| 71 | D4 | (1−24T+4pT2−24pT3+p2T4)2 |
| 73 | D4×C2 | 1−130T2+12291T4−130p2T6+p4T8 |
| 79 | C4 | (1−4T+90T2−4pT3+p2T4)2 |
| 83 | D4×C2 | 1−116T2+6774T4−116p2T6+p4T8 |
| 89 | C22 | (1+128T2+p2T4)2 |
| 97 | D4×C2 | 1−252T2+30086T4−252p2T6+p4T8 |
show more | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.70060184641792138131439715967, −6.37148812261193787453585518580, −6.32502142692066894376224366107, −6.26713134262306995986363608635, −6.24842408461878396296566214830, −5.38861425307692806763428886269, −5.28535667006082211774502065414, −5.24308630160730822218457559836, −5.19670382211094827126225664815, −4.69878047542539799213847256189, −4.64259857386581524762749208680, −4.01013335217070721917736809449, −3.95450526932887332602264172729, −3.92699870702487615962091440827, −3.83994115418674686283762026751, −3.46681132702447006265933140103, −3.06253829825956769926206886081, −2.62845215827761559126014506746, −2.29581768583533008892180805253, −2.25388253093288475349575886833, −2.00809282202571200259223964157, −1.53091810429916511590236429102, −1.24957277443162886590991482531, −0.74638202948797869613309201616, −0.53058713385779782675361403418,
0.53058713385779782675361403418, 0.74638202948797869613309201616, 1.24957277443162886590991482531, 1.53091810429916511590236429102, 2.00809282202571200259223964157, 2.25388253093288475349575886833, 2.29581768583533008892180805253, 2.62845215827761559126014506746, 3.06253829825956769926206886081, 3.46681132702447006265933140103, 3.83994115418674686283762026751, 3.92699870702487615962091440827, 3.95450526932887332602264172729, 4.01013335217070721917736809449, 4.64259857386581524762749208680, 4.69878047542539799213847256189, 5.19670382211094827126225664815, 5.24308630160730822218457559836, 5.28535667006082211774502065414, 5.38861425307692806763428886269, 6.24842408461878396296566214830, 6.26713134262306995986363608635, 6.32502142692066894376224366107, 6.37148812261193787453585518580, 6.70060184641792138131439715967