L(s) = 1 | + 4·7-s + 20·13-s + 28·19-s − 10·25-s − 8·31-s + 20·37-s − 152·43-s − 186·49-s − 100·61-s − 284·67-s + 164·73-s + 76·79-s + 80·91-s + 20·97-s + 100·103-s + 248·109-s + 124·121-s + 127-s + 131-s + 112·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 4/7·7-s + 1.53·13-s + 1.47·19-s − 2/5·25-s − 0.258·31-s + 0.540·37-s − 3.53·43-s − 3.79·49-s − 1.63·61-s − 4.23·67-s + 2.24·73-s + 0.962·79-s + 0.879·91-s + 0.206·97-s + 0.970·103-s + 2.27·109-s + 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.842·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s)4L(s)Λ(3−s)
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s+1)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅312⋅54
|
Sign: |
1
|
Analytic conductor: |
1.19992×107 |
Root analytic conductor: |
7.67174 |
Motivic weight: |
2 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅312⋅54, ( :1,1,1,1), 1)
|
Particular Values
L(23) |
≈ |
4.753758070 |
L(21) |
≈ |
4.753758070 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C2 | (1+pT2)2 |
good | 7 | C2 | (1−T+p2T2)4 |
| 11 | C22 | (1−62T2+p4T4)2 |
| 13 | D4 | (1−10T+3T2−10p2T3+p4T4)2 |
| 17 | C22 | (1−506T2+p4T4)2 |
| 19 | D4 | (1−14T+411T2−14p2T3+p4T4)2 |
| 23 | D4×C2 | 1−1180T2+700422T4−1180p4T6+p8T8 |
| 29 | D4×C2 | 1−2860T2+3407622T4−2860p4T6+p8T8 |
| 31 | D4 | (1+4T+486T2+4p2T3+p4T4)2 |
| 37 | D4 | (1−10T+2403T2−10p2T3+p4T4)2 |
| 41 | D4×C2 | 1−2764T2+6265446T4−2764p4T6+p8T8 |
| 43 | D4 | (1+76T+3702T2+76p2T3+p4T4)2 |
| 47 | D4×C2 | 1−3796T2+8177766T4−3796p4T6+p8T8 |
| 53 | D4×C2 | 1+788T2+11737158T4+788p4T6+p8T8 |
| 59 | C22 | (1−6890T2+p4T4)2 |
| 61 | D4 | (1+50T+2307T2+50p2T3+p4T4)2 |
| 67 | C2 | (1+71T+p2T2)4 |
| 71 | D4×C2 | 1−4108T2−8461722T4−4108p4T6+p8T8 |
| 73 | D4 | (1−82T+11979T2−82p2T3+p4T4)2 |
| 79 | D4 | (1−38T+3843T2−38p2T3+p4T4)2 |
| 83 | D4×C2 | 1−2860T2−27506298T4−2860p4T6+p8T8 |
| 89 | D4×C2 | 1−8500T2+43184742T4−8500p4T6+p8T8 |
| 97 | D4 | (1−10T+9843T2−10p2T3+p4T4)2 |
show more | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.18495232679088504739467416329, −5.99156223746308432532786381251, −5.83862117625269490456209145751, −5.82222448265045612275375802269, −5.47458284324530327712570423046, −5.07760150052092594302480328106, −4.86137262699263793778896126981, −4.81187294954066147512841215887, −4.58327656710794239703475110220, −4.56422648896260737249973640007, −4.16406156022060616963289090126, −3.63281774689853563112723486826, −3.51742181365757417959803786801, −3.49345088315036323867723304509, −3.19860559882417072574132527397, −2.96364132159082060686103663904, −2.94984265181886267038961321384, −2.23417507735417806680388901924, −2.00586954426064664645434456763, −1.70488372864534811606911064339, −1.64677964398243202243299864318, −1.25705081487685245605574828306, −1.10502704315814740436573323873, −0.42581816998833457435218443018, −0.34844598642026516687069602614,
0.34844598642026516687069602614, 0.42581816998833457435218443018, 1.10502704315814740436573323873, 1.25705081487685245605574828306, 1.64677964398243202243299864318, 1.70488372864534811606911064339, 2.00586954426064664645434456763, 2.23417507735417806680388901924, 2.94984265181886267038961321384, 2.96364132159082060686103663904, 3.19860559882417072574132527397, 3.49345088315036323867723304509, 3.51742181365757417959803786801, 3.63281774689853563112723486826, 4.16406156022060616963289090126, 4.56422648896260737249973640007, 4.58327656710794239703475110220, 4.81187294954066147512841215887, 4.86137262699263793778896126981, 5.07760150052092594302480328106, 5.47458284324530327712570423046, 5.82222448265045612275375802269, 5.83862117625269490456209145751, 5.99156223746308432532786381251, 6.18495232679088504739467416329