L(s) = 1 | − 58i·5-s + 2.18e3·9-s + 8.89e3i·13-s − 4.00e4·17-s + 7.47e4·25-s − 2.33e5i·29-s + 5.63e5i·37-s − 9.53e3·41-s − 1.26e5i·45-s − 8.23e5·49-s − 7.98e5i·53-s + 3.50e6i·61-s + 5.16e5·65-s − 3.91e6·73-s + 4.78e6·81-s + ⋯ |
L(s) = 1 | − 0.207i·5-s + 9-s + 1.12i·13-s − 1.97·17-s + 0.956·25-s − 1.77i·29-s + 1.83i·37-s − 0.0215·41-s − 0.207i·45-s − 49-s − 0.736i·53-s + 1.97i·61-s + 0.233·65-s − 1.17·73-s + 81-s + ⋯ |
Λ(s)=(=(256s/2ΓC(s)L(s)(−0.707−0.707i)Λ(8−s)
Λ(s)=(=(256s/2ΓC(s+7/2)L(s)(−0.707−0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
256
= 28
|
Sign: |
−0.707−0.707i
|
Analytic conductor: |
79.9705 |
Root analytic conductor: |
8.94262 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ256(129,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 256, ( :7/2), −0.707−0.707i)
|
Particular Values
L(4) |
≈ |
0.8588077309 |
L(21) |
≈ |
0.8588077309 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
good | 3 | 1−2.18e3T2 |
| 5 | 1+58iT−7.81e4T2 |
| 7 | 1+8.23e5T2 |
| 11 | 1−1.94e7T2 |
| 13 | 1−8.89e3iT−6.27e7T2 |
| 17 | 1+4.00e4T+4.10e8T2 |
| 19 | 1−8.93e8T2 |
| 23 | 1+3.40e9T2 |
| 29 | 1+2.33e5iT−1.72e10T2 |
| 31 | 1+2.75e10T2 |
| 37 | 1−5.63e5iT−9.49e10T2 |
| 41 | 1+9.53e3T+1.94e11T2 |
| 43 | 1−2.71e11T2 |
| 47 | 1+5.06e11T2 |
| 53 | 1+7.98e5iT−1.17e12T2 |
| 59 | 1−2.48e12T2 |
| 61 | 1−3.50e6iT−3.14e12T2 |
| 67 | 1−6.06e12T2 |
| 71 | 1+9.09e12T2 |
| 73 | 1+3.91e6T+1.10e13T2 |
| 79 | 1+1.92e13T2 |
| 83 | 1−2.71e13T2 |
| 89 | 1+9.24e6T+4.42e13T2 |
| 97 | 1+1.75e7T+8.07e13T2 |
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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
−11.20255807738571982125331653307, −10.10120348455262132812530448503, −9.229796821030971709167218716401, −8.323826884162768501970937251353, −7.01468541366574405450022340293, −6.38757839404935632178326071947, −4.73182339802749230894809410282, −4.14090996317143924029891382775, −2.43809156404570826930909614831, −1.32489436017951333899372613098,
0.19462660925242371691595585269, 1.59889341212149281161137643746, 2.90568590634306187181472596953, 4.19144476908664730311459225371, 5.22514309247016007453366243162, 6.59003793280927384686564897775, 7.31380105152670964941081467558, 8.527168978966713563428536029372, 9.441281789637447849265232830041, 10.62613301583465156320378916628