L(s) = 1 | + (−0.793 − 1.17i)2-s + (1.68 + 0.898i)3-s + (−0.742 + 1.85i)4-s + (−1.56 + 1.91i)5-s + (−0.281 − 2.68i)6-s + (3.63 + 2.43i)7-s + (2.76 − 0.603i)8-s + (0.353 + 0.529i)9-s + (3.48 + 0.321i)10-s + (−4.39 − 1.33i)11-s + (−2.91 + 2.45i)12-s + (1.74 − 1.42i)13-s + (−0.0388 − 6.19i)14-s + (−4.35 + 1.80i)15-s + (−2.89 − 2.75i)16-s + (3.75 + 1.55i)17-s + ⋯ |
L(s) = 1 | + (−0.560 − 0.827i)2-s + (0.970 + 0.518i)3-s + (−0.371 + 0.928i)4-s + (−0.701 + 0.854i)5-s + (−0.114 − 1.09i)6-s + (1.37 + 0.919i)7-s + (0.976 − 0.213i)8-s + (0.117 + 0.176i)9-s + (1.10 + 0.101i)10-s + (−1.32 − 0.401i)11-s + (−0.842 + 0.709i)12-s + (0.483 − 0.396i)13-s + (−0.0103 − 1.65i)14-s + (−1.12 + 0.466i)15-s + (−0.724 − 0.689i)16-s + (0.909 + 0.376i)17-s + ⋯ |
Λ(s)=(=(128s/2ΓC(s)L(s)(0.981−0.189i)Λ(2−s)
Λ(s)=(=(128s/2ΓC(s+1/2)L(s)(0.981−0.189i)Λ(1−s)
Degree: |
2 |
Conductor: |
128
= 27
|
Sign: |
0.981−0.189i
|
Analytic conductor: |
1.02208 |
Root analytic conductor: |
1.01098 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ128(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 128, ( :1/2), 0.981−0.189i)
|
Particular Values
L(1) |
≈ |
1.02299+0.0980705i |
L(21) |
≈ |
1.02299+0.0980705i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.793+1.17i)T |
good | 3 | 1+(−1.68−0.898i)T+(1.66+2.49i)T2 |
| 5 | 1+(1.56−1.91i)T+(−0.975−4.90i)T2 |
| 7 | 1+(−3.63−2.43i)T+(2.67+6.46i)T2 |
| 11 | 1+(4.39+1.33i)T+(9.14+6.11i)T2 |
| 13 | 1+(−1.74+1.42i)T+(2.53−12.7i)T2 |
| 17 | 1+(−3.75−1.55i)T+(12.0+12.0i)T2 |
| 19 | 1+(−0.804+8.16i)T+(−18.6−3.70i)T2 |
| 23 | 1+(1.74+0.347i)T+(21.2+8.80i)T2 |
| 29 | 1+(−0.598−1.97i)T+(−24.1+16.1i)T2 |
| 31 | 1+(3.81−3.81i)T−31iT2 |
| 37 | 1+(1.04−0.102i)T+(36.2−7.21i)T2 |
| 41 | 1+(−0.711+3.57i)T+(−37.8−15.6i)T2 |
| 43 | 1+(0.793−0.423i)T+(23.8−35.7i)T2 |
| 47 | 1+(−1.43+3.46i)T+(−33.2−33.2i)T2 |
| 53 | 1+(−1.79+5.91i)T+(−44.0−29.4i)T2 |
| 59 | 1+(4.26+3.49i)T+(11.5+57.8i)T2 |
| 61 | 1+(2.89−5.41i)T+(−33.8−50.7i)T2 |
| 67 | 1+(2.61−4.89i)T+(−37.2−55.7i)T2 |
| 71 | 1+(−0.916+1.37i)T+(−27.1−65.5i)T2 |
| 73 | 1+(−6.52+4.36i)T+(27.9−67.4i)T2 |
| 79 | 1+(4.05+9.80i)T+(−55.8+55.8i)T2 |
| 83 | 1+(0.630+0.0621i)T+(81.4+16.1i)T2 |
| 89 | 1+(−3.26+0.649i)T+(82.2−34.0i)T2 |
| 97 | 1+(12.6−12.6i)T−97iT2 |
show more | |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.38296094840089922269989269265, −12.03027316090919291868381780690, −11.11111147183805602827016805586, −10.47816478386126404860251873630, −9.006578995458521407068255385024, −8.301739592640086702681419332310, −7.54490164628953767390668762033, −5.09211553610704047339388490087, −3.45210484051890464526484260178, −2.54110431857810722440460355564,
1.54360325812182180933004949612, 4.24003204819980336188134041832, 5.43741241230721573118911457962, 7.66199479188589657737530140411, 7.76869991972290422110601982939, 8.507370103133028687115291104304, 9.959551090229819163050929908092, 11.08739180895677751576310876865, 12.51954024286557126994115700370, 13.76355045996499708643853754561