L(s) = 1 | + 4·13-s − 2·25-s − 28·37-s + 22·49-s − 4·61-s + 4·73-s + 4·97-s + 40·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 1.10·13-s − 2/5·25-s − 4.60·37-s + 22/7·49-s − 0.512·61-s + 0.468·73-s + 0.406·97-s + 3.83·109-s − 1.81·121-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 − 3.23·169-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 + 0.0688·211-s + ⋯ |
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅312⋅54
|
Sign: |
1
|
Analytic conductor: |
88495.9 |
Root analytic conductor: |
4.15303 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.5706442043 |
L(21) |
≈ |
0.5706442043 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C2 | (1+T2)2 |
good | 7 | C2 | (1−5T+pT2)2(1+5T+pT2)2 |
| 11 | C22 | (1+10T2+p2T4)2 |
| 13 | C2 | (1−T+pT2)4 |
| 17 | C2 | (1−pT2)4 |
| 19 | C22 | (1−35T2+p2T4)2 |
| 23 | C22 | (1+34T2+p2T4)2 |
| 29 | C22 | (1−22T2+p2T4)2 |
| 31 | C22 | (1−50T2+p2T4)2 |
| 37 | C2 | (1+7T+pT2)4 |
| 41 | C22 | (1−46T2+p2T4)2 |
| 43 | C22 | (1−74T2+p2T4)2 |
| 47 | C22 | (1+46T2+p2T4)2 |
| 53 | C22 | (1−70T2+p2T4)2 |
| 59 | C22 | (1−74T2+p2T4)2 |
| 61 | C2 | (1+T+pT2)4 |
| 67 | C22 | (1−107T2+p2T4)2 |
| 71 | C22 | (1+130T2+p2T4)2 |
| 73 | C2 | (1−T+pT2)4 |
| 79 | C22 | (1−83T2+p2T4)2 |
| 83 | C22 | (1+58T2+p2T4)2 |
| 89 | C2 | (1−pT2)4 |
| 97 | C2 | (1−T+pT2)4 |
show more | | |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.33099047397905100550774477181, −6.23624482994640978458263359942, −5.99532774583619869436576379465, −5.94745949388314923772472107160, −5.52906071721795402969418661162, −5.47426832600293440783023661910, −5.10411608875379806072954311305, −5.07804056398547159204739192426, −4.81598241870371231887458899593, −4.58399955471075539385164287439, −4.24107494207845485717800315905, −3.94408378957739154941273383026, −3.85203247271678446037838200895, −3.54559552449163762711873395931, −3.49633787112694568590308516523, −3.27462571662969934193773560785, −2.77045584621031606995033385689, −2.73080153803482116321796042247, −2.14178722861798386979094389402, −2.13694360238549647192676374937, −1.75796999180877063508061669009, −1.54165185789996133310112757407, −0.963429626694168764341601162701, −0.937952479543584697810901185521, −0.12976108285368816478317225020,
0.12976108285368816478317225020, 0.937952479543584697810901185521, 0.963429626694168764341601162701, 1.54165185789996133310112757407, 1.75796999180877063508061669009, 2.13694360238549647192676374937, 2.14178722861798386979094389402, 2.73080153803482116321796042247, 2.77045584621031606995033385689, 3.27462571662969934193773560785, 3.49633787112694568590308516523, 3.54559552449163762711873395931, 3.85203247271678446037838200895, 3.94408378957739154941273383026, 4.24107494207845485717800315905, 4.58399955471075539385164287439, 4.81598241870371231887458899593, 5.07804056398547159204739192426, 5.10411608875379806072954311305, 5.47426832600293440783023661910, 5.52906071721795402969418661162, 5.94745949388314923772472107160, 5.99532774583619869436576379465, 6.23624482994640978458263359942, 6.33099047397905100550774477181