L(s) = 1 | + 4·3-s − 4·5-s + 4·7-s + 8·9-s − 2·13-s − 16·15-s − 10·17-s + 8·19-s + 16·21-s + 4·23-s + 11·25-s + 12·27-s − 16·35-s + 2·37-s − 8·39-s − 12·43-s − 32·45-s − 4·47-s + 8·49-s − 40·51-s − 14·53-s + 32·57-s + 8·59-s − 8·61-s + 32·63-s + 8·65-s − 20·67-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 1.78·5-s + 1.51·7-s + 8/3·9-s − 0.554·13-s − 4.13·15-s − 2.42·17-s + 1.83·19-s + 3.49·21-s + 0.834·23-s + 11/5·25-s + 2.30·27-s − 2.70·35-s + 0.328·37-s − 1.28·39-s − 1.82·43-s − 4.77·45-s − 0.583·47-s + 8/7·49-s − 5.60·51-s − 1.92·53-s + 4.23·57-s + 1.04·59-s − 1.02·61-s + 4.03·63-s + 0.992·65-s − 2.44·67-s + ⋯ |
Λ(s)=(=(25600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(25600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
25600
= 210⋅52
|
Sign: |
1
|
Analytic conductor: |
1.63227 |
Root analytic conductor: |
1.13031 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 25600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.079274822 |
L(21) |
≈ |
2.079274822 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C2 | 1+4T+pT2 |
good | 3 | C22 | 1−4T+8T2−4pT3+p2T4 |
| 7 | C22 | 1−4T+8T2−4pT3+p2T4 |
| 11 | C2 | (1−pT2)2 |
| 13 | C2 | (1−4T+pT2)(1+6T+pT2) |
| 17 | C2 | (1+2T+pT2)(1+8T+pT2) |
| 19 | C2 | (1−4T+pT2)2 |
| 23 | C22 | 1−4T+8T2−4pT3+p2T4 |
| 29 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 31 | C22 | 1−46T2+p2T4 |
| 37 | C22 | 1−2T+2T2−2pT3+p2T4 |
| 41 | C2 | (1+pT2)2 |
| 43 | C22 | 1+12T+72T2+12pT3+p2T4 |
| 47 | C22 | 1+4T+8T2+4pT3+p2T4 |
| 53 | C22 | 1+14T+98T2+14pT3+p2T4 |
| 59 | C2 | (1−4T+pT2)2 |
| 61 | C2 | (1+4T+pT2)2 |
| 67 | C22 | 1+20T+200T2+20pT3+p2T4 |
| 71 | C22 | 1+2T2+p2T4 |
| 73 | C22 | 1+6T+18T2+6pT3+p2T4 |
| 79 | C2 | (1−16T+pT2)2 |
| 83 | C22 | 1+4T+8T2+4pT3+p2T4 |
| 89 | C2 | (1−pT2)2 |
| 97 | C22 | 1+6T+18T2+6pT3+p2T4 |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.52948900660978997700739454452, −12.71374801099312588499214377472, −12.07023270002773118411229294502, −11.67769579780215213176866036881, −11.02035611823193949200370460922, −10.98230605233472102033642787198, −9.939661053809370859920349707089, −9.126987559540030977544371724928, −9.010490516613839272564350717719, −8.413988397954181114823838919408, −7.929833937490584685691239283390, −7.79759963680671264523446257947, −7.14425004370225237975692630081, −6.65685970773035818937017623377, −4.89861537428938523849156835685, −4.80279718484695034016716814816, −4.04291414657135911424569089257, −3.24165158360942039221965694743, −2.79192753768080197357103162731, −1.72813720098024007817035789707,
1.72813720098024007817035789707, 2.79192753768080197357103162731, 3.24165158360942039221965694743, 4.04291414657135911424569089257, 4.80279718484695034016716814816, 4.89861537428938523849156835685, 6.65685970773035818937017623377, 7.14425004370225237975692630081, 7.79759963680671264523446257947, 7.929833937490584685691239283390, 8.413988397954181114823838919408, 9.010490516613839272564350717719, 9.126987559540030977544371724928, 9.939661053809370859920349707089, 10.98230605233472102033642787198, 11.02035611823193949200370460922, 11.67769579780215213176866036881, 12.07023270002773118411229294502, 12.71374801099312588499214377472, 13.52948900660978997700739454452