L(s) = 1 | + 512·2-s + 1.76e4·3-s + 1.96e5·4-s − 7.81e5·5-s + 9.02e6·6-s + 2.76e7·7-s + 6.71e7·8-s + 1.44e8·9-s − 4.00e8·10-s + 6.45e7·11-s + 3.46e9·12-s + 2.89e9·13-s + 1.41e10·14-s − 1.37e10·15-s + 2.14e10·16-s + 1.58e9·17-s + 7.38e10·18-s + 4.42e10·19-s − 1.53e11·20-s + 4.88e11·21-s + 3.30e10·22-s + 4.87e11·23-s + 1.18e12·24-s + 4.57e11·25-s + 1.48e12·26-s + 1.88e12·27-s + 5.44e12·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.55·3-s + 3/2·4-s − 0.894·5-s + 2.19·6-s + 1.81·7-s + 1.41·8-s + 1.11·9-s − 1.26·10-s + 0.0907·11-s + 2.32·12-s + 0.984·13-s + 2.56·14-s − 1.38·15-s + 5/4·16-s + 0.0549·17-s + 1.57·18-s + 0.597·19-s − 1.34·20-s + 2.81·21-s + 0.128·22-s + 1.29·23-s + 2.19·24-s + 3/5·25-s + 1.39·26-s + 1.28·27-s + 2.72·28-s + ⋯ |
Λ(s)=(=(100s/2ΓC(s)2L(s)Λ(18−s)
Λ(s)=(=(100s/2ΓC(s+17/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
100
= 22⋅52
|
Sign: |
1
|
Analytic conductor: |
335.703 |
Root analytic conductor: |
4.28044 |
Motivic weight: |
17 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 100, ( :17/2,17/2), 1)
|
Particular Values
L(9) |
≈ |
14.70281786 |
L(21) |
≈ |
14.70281786 |
L(219) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−p8T)2 |
| 5 | C1 | (1+p8T)2 |
good | 3 | D4 | 1−5876pT+685454p5T2−5876p18T3+p34T4 |
| 7 | D4 | 1−27684196T+93506538073374pT2−27684196p17T3+p34T4 |
| 11 | D4 | 1−533184p2T−3200366345168954p2T2−533184p19T3+p34T4 |
| 13 | D4 | 1−2895838468T+1418824834264307094pT2−2895838468p17T3+p34T4 |
| 17 | D4 | 1−1580212596T+13⋯58T2−1580212596p17T3+p34T4 |
| 19 | D4 | 1−44213712760T+27⋯78T2−44213712760p17T3+p34T4 |
| 23 | D4 | 1−487549782828T+24⋯02T2−487549782828p17T3+p34T4 |
| 29 | D4 | 1−3987314863500T+18⋯18T2−3987314863500p17T3+p34T4 |
| 31 | D4 | 1+5492261339336T+52⋯46T2+5492261339336p17T3+p34T4 |
| 37 | D4 | 1+62715287637884T+18⋯98T2+62715287637884p17T3+p34T4 |
| 41 | D4 | 1−23411477277324T+10⋯06T2−23411477277324p17T3+p34T4 |
| 43 | D4 | 1+124856923191092T+14⋯02T2+124856923191092p17T3+p34T4 |
| 47 | D4 | 1+185946612123564T+57⋯98T2+185946612123564p17T3+p34T4 |
| 53 | D4 | 1−359339780647668T+33⋯82T2−359339780647668p17T3+p34T4 |
| 59 | D4 | 1−902179170360600T+20⋯38T2−902179170360600p17T3+p34T4 |
| 61 | D4 | 1+1564422918967676T+36⋯86T2+1564422918967676p17T3+p34T4 |
| 67 | D4 | 1+5839738931054684T+30⋯18T2+5839738931054684p17T3+p34T4 |
| 71 | D4 | 1+67588560434136T+50⋯06T2+67588560434136p17T3+p34T4 |
| 73 | D4 | 1−3533390699585668T+90⋯62T2−3533390699585668p17T3+p34T4 |
| 79 | D4 | 1−19002656396552080T+43⋯18T2−19002656396552080p17T3+p34T4 |
| 83 | D4 | 1−261145638254436pT+75⋯82T2−261145638254436p18T3+p34T4 |
| 89 | D4 | 1+97499522192222220T+48⋯58T2+97499522192222220p17T3+p34T4 |
| 97 | D4 | 1−99889937855386756T+14⋯58T2−99889937855386756p17T3+p34T4 |
show more | | |
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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
−16.49452189855450298359084947718, −15.79977185819673406160331055604, −14.91810322526225841113093940138, −14.89257222253528925539165822730, −13.87900508727161083944734268257, −13.82863554692487995426831217217, −12.66806956792330355632198207896, −11.85444941196240143799885967530, −11.21093525111772113575974931888, −10.48194131361407213808657194346, −8.716239185577053457074559797551, −8.414702524300610678224362229570, −7.56453926402556833448452837594, −6.76098960845187442525068610138, −5.11636671786388053697245566920, −4.66794664716345343582458429224, −3.48709245950675540928065591627, −3.17904426674912031285594886645, −1.90031851112469128941490957302, −1.22162989460489883800798337344,
1.22162989460489883800798337344, 1.90031851112469128941490957302, 3.17904426674912031285594886645, 3.48709245950675540928065591627, 4.66794664716345343582458429224, 5.11636671786388053697245566920, 6.76098960845187442525068610138, 7.56453926402556833448452837594, 8.414702524300610678224362229570, 8.716239185577053457074559797551, 10.48194131361407213808657194346, 11.21093525111772113575974931888, 11.85444941196240143799885967530, 12.66806956792330355632198207896, 13.82863554692487995426831217217, 13.87900508727161083944734268257, 14.89257222253528925539165822730, 14.91810322526225841113093940138, 15.79977185819673406160331055604, 16.49452189855450298359084947718