L(s) = 1 | + 2.04e3·2-s + 3.09e4·3-s + 3.14e6·4-s − 1.95e7·5-s + 6.34e7·6-s − 4.39e8·7-s + 4.29e9·8-s − 3.21e9·9-s − 4.00e10·10-s + 1.05e11·11-s + 9.74e10·12-s + 3.08e11·13-s − 9.01e11·14-s − 6.04e11·15-s + 5.49e12·16-s + 1.83e13·17-s − 6.58e12·18-s + 2.38e13·19-s − 6.14e13·20-s − 1.36e13·21-s + 2.15e14·22-s − 8.25e13·23-s + 1.33e14·24-s + 2.86e14·25-s + 6.31e14·26-s + 9.51e13·27-s − 1.38e15·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.302·3-s + 3/2·4-s − 0.894·5-s + 0.428·6-s − 0.588·7-s + 1.41·8-s − 0.307·9-s − 1.26·10-s + 1.22·11-s + 0.454·12-s + 0.620·13-s − 0.832·14-s − 0.270·15-s + 5/4·16-s + 2.21·17-s − 0.434·18-s + 0.891·19-s − 1.34·20-s − 0.178·21-s + 1.72·22-s − 0.415·23-s + 0.428·24-s + 3/5·25-s + 0.877·26-s + 0.0889·27-s − 0.883·28-s + ⋯ |
Λ(s)=(=(100s/2ΓC(s)2L(s)Λ(22−s)
Λ(s)=(=(100s/2ΓC(s+21/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
100
= 22⋅52
|
Sign: |
1
|
Analytic conductor: |
781.075 |
Root analytic conductor: |
5.28656 |
Motivic weight: |
21 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 100, ( :21/2,21/2), 1)
|
Particular Values
L(11) |
≈ |
8.482990907 |
L(21) |
≈ |
8.482990907 |
L(223) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−p10T)2 |
| 5 | C1 | (1+p10T)2 |
good | 3 | D4 | 1−10324pT+17175214p5T2−10324p22T3+p42T4 |
| 7 | D4 | 1+439959356T+166460827342795614pT2+439959356p21T3+p42T4 |
| 11 | D4 | 1−105191777184T+74⋯26pT2−105191777184p21T3+p42T4 |
| 13 | D4 | 1−308456648932T+31⋯14pT2−308456648932p21T3+p42T4 |
| 17 | D4 | 1−1081999770612pT+72⋯42p2T2−1081999770612p22T3+p42T4 |
| 19 | D4 | 1−1253748248200pT+28⋯58p2T2−1253748248200p22T3+p42T4 |
| 23 | D4 | 1+82586042978868T+77⋯02T2+82586042978868p21T3+p42T4 |
| 29 | D4 | 1+167038888446420T+27⋯58T2+167038888446420p21T3+p42T4 |
| 31 | D4 | 1−5373084998145784T+46⋯26T2−5373084998145784p21T3+p42T4 |
| 37 | D4 | 1−83725084127912164T+32⋯98T2−83725084127912164p21T3+p42T4 |
| 41 | D4 | 1+63222375005514036T+15⋯06T2+63222375005514036p21T3+p42T4 |
| 43 | D4 | 1+113835841911164948T+12⋯62T2+113835841911164948p21T3+p42T4 |
| 47 | D4 | 1−13253549001226164T+24⋯18T2−13253549001226164p21T3+p42T4 |
| 53 | D4 | 1+1445751023743904748T+67⋯82T2+1445751023743904748p21T3+p42T4 |
| 59 | D4 | 1+817060118931432840T+30⋯18T2+817060118931432840p21T3+p42T4 |
| 61 | D4 | 1+4580997169825849436T+32⋯46T2+4580997169825849436p21T3+p42T4 |
| 67 | D4 | 1+28808031668773210556T+64⋯18T2+28808031668773210556p21T3+p42T4 |
| 71 | D4 | 1−51140957732016114984T+21⋯06T2−51140957732016114984p21T3+p42T4 |
| 73 | D4 | 1−14792322512082321412T+38⋯82T2−14792322512082321412p21T3+p42T4 |
| 79 | D4 | 1−25077766877525032720T+11⋯58T2−25077766877525032720p21T3+p42T4 |
| 83 | D4 | 1+12⋯68T+38⋯22T2+12⋯68p21T3+p42T4 |
| 89 | D4 | 1−10⋯80T−33⋯22T2−10⋯80p21T3+p42T4 |
| 97 | D4 | 1+26⋯96T+45⋯98T2+26⋯96p21T3+p42T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−15.80904391050943419199518934072, −15.13882234462509989118536923004, −14.30191180182851090555111216750, −14.16960218827182429796380241887, −13.11501084736210432123292800820, −12.44505953192757271287888041570, −11.61423277013174624547472955210, −11.53921826035046864923926285742, −10.16505969652227915066826270464, −9.426914530971558626016586784040, −8.100065386988730858243015776923, −7.63250874819259180382847280239, −6.45860842729809545368635697469, −6.01437335541878318939133969791, −4.89552429151551962596653638970, −4.05255449788773471739179722733, −3.20267462228395308125262242873, −3.10198697336722944316667572319, −1.48514684380487484315958066096, −0.794842427459680101765561788939,
0.794842427459680101765561788939, 1.48514684380487484315958066096, 3.10198697336722944316667572319, 3.20267462228395308125262242873, 4.05255449788773471739179722733, 4.89552429151551962596653638970, 6.01437335541878318939133969791, 6.45860842729809545368635697469, 7.63250874819259180382847280239, 8.100065386988730858243015776923, 9.426914530971558626016586784040, 10.16505969652227915066826270464, 11.53921826035046864923926285742, 11.61423277013174624547472955210, 12.44505953192757271287888041570, 13.11501084736210432123292800820, 14.16960218827182429796380241887, 14.30191180182851090555111216750, 15.13882234462509989118536923004, 15.80904391050943419199518934072