L(s) = 1 | − 7.68e3·2-s − 2.01e7·4-s + 9.76e8·5-s + 2.49e10·7-s + 1.65e11·8-s − 7.50e12·10-s − 9.98e12·11-s − 1.43e14·13-s − 1.91e14·14-s + 1.23e15·16-s − 2.80e15·17-s + 1.50e16·19-s − 1.97e16·20-s + 7.67e16·22-s − 4.87e16·23-s + 5.96e17·25-s + 1.10e18·26-s − 5.02e17·28-s + 3.66e17·29-s + 5.26e18·31-s − 2.14e18·32-s + 2.15e19·34-s + 2.43e19·35-s − 8.25e19·37-s − 1.15e20·38-s + 1.61e20·40-s − 1.16e20·41-s + ⋯ |
L(s) = 1 | − 1.32·2-s − 0.601·4-s + 1.78·5-s + 0.680·7-s + 0.850·8-s − 2.37·10-s − 0.959·11-s − 1.70·13-s − 0.902·14-s + 1.09·16-s − 1.16·17-s + 1.55·19-s − 1.07·20-s + 1.27·22-s − 0.464·23-s + 2·25-s + 2.26·26-s − 0.409·28-s + 0.192·29-s + 1.20·31-s − 0.328·32-s + 1.54·34-s + 1.21·35-s − 2.06·37-s − 2.06·38-s + 1.52·40-s − 0.804·41-s + ⋯ |
Λ(s)=(=(4100625s/2ΓC(s)4L(s)Λ(26−s)
Λ(s)=(=(4100625s/2ΓC(s+25/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
4100625
= 38⋅54
|
Sign: |
1
|
Analytic conductor: |
1.00836×109 |
Root analytic conductor: |
13.3491 |
Motivic weight: |
25 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 4100625, ( :25/2,25/2,25/2,25/2), 1)
|
Particular Values
L(13) |
= |
0 |
L(21) |
= |
0 |
L(227) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.261778601273865218635055797278, −7.55763149504390845720637833992, −7.40242982521219263200210055907, −7.00798891434045335509220859301, −6.78349234785790289689252463220, −6.50432857365175385161024787479, −5.98317206398814351064296627023, −5.93073559791205231896715917495, −5.36407160517188913990754188299, −5.10883254847517096989700727407, −5.03932731376465296767391536262, −4.89466557570648250987038425641, −4.70116037586858864969836496570, −3.88407381251661718001439524430, −3.73450594082398097344142526678, −3.61906667808783979400479432925, −2.84110472395185403962082277847, −2.66676425648311612699477196126, −2.47671670759645185353059708988, −2.24755251728973214491999018172, −2.07097706028519140559650085514, −1.46856057943459527341415855780, −1.10866205417874722541528976439, −1.10520762670988228496773997418, −0.894192834311552323981584345390, 0, 0, 0, 0,
0.894192834311552323981584345390, 1.10520762670988228496773997418, 1.10866205417874722541528976439, 1.46856057943459527341415855780, 2.07097706028519140559650085514, 2.24755251728973214491999018172, 2.47671670759645185353059708988, 2.66676425648311612699477196126, 2.84110472395185403962082277847, 3.61906667808783979400479432925, 3.73450594082398097344142526678, 3.88407381251661718001439524430, 4.70116037586858864969836496570, 4.89466557570648250987038425641, 5.03932731376465296767391536262, 5.10883254847517096989700727407, 5.36407160517188913990754188299, 5.93073559791205231896715917495, 5.98317206398814351064296627023, 6.50432857365175385161024787479, 6.78349234785790289689252463220, 7.00798891434045335509220859301, 7.40242982521219263200210055907, 7.55763149504390845720637833992, 8.261778601273865218635055797278