L(s) = 1 | + 176·25-s + 1.37e3·49-s − 2.36e3·73-s − 7.26e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 1.40·25-s + 4·49-s − 3.79·73-s − 7.60·97-s + 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
Λ(s)=(=((228⋅38)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((228⋅38)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
228⋅38
|
Sign: |
1
|
Analytic conductor: |
2.13439×107 |
Root analytic conductor: |
8.24440 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 228⋅38, ( :3/2,3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
0.3219958424 |
L(21) |
≈ |
0.3219958424 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
good | 5 | C22 | (1−88T2+p6T4)2 |
| 7 | C2 | (1−p3T2)4 |
| 11 | C2 | (1−p3T2)4 |
| 13 | C2 | (1−92T+p3T2)2(1+92T+p3T2)2 |
| 17 | C22 | (1+9776T2+p6T4)2 |
| 19 | C2 | (1+p3T2)4 |
| 23 | C2 | (1+p3T2)4 |
| 29 | C22 | (1+36920T2+p6T4)2 |
| 31 | C2 | (1−p3T2)4 |
| 37 | C2 | (1−214T+p3T2)2(1+214T+p3T2)2 |
| 41 | C22 | (1+108560T2+p6T4)2 |
| 43 | C2 | (1+p3T2)4 |
| 47 | C2 | (1+p3T2)4 |
| 53 | C22 | (1−296296T2+p6T4)2 |
| 59 | C2 | (1−p3T2)4 |
| 61 | C2 | (1−830T+p3T2)2(1+830T+p3T2)2 |
| 67 | C2 | (1+p3T2)4 |
| 71 | C2 | (1+p3T2)4 |
| 73 | C2 | (1+592T+p3T2)4 |
| 79 | C2 | (1−p3T2)4 |
| 83 | C2 | (1−p3T2)4 |
| 89 | C22 | (1−293920T2+p6T4)2 |
| 97 | C2 | (1+1816T+p3T2)4 |
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
−6.73422977769241169477222487143, −6.61893424048649174697038790167, −6.05287804043078955779494127528, −5.81770935679324149884641554258, −5.65507343447641795673971559459, −5.55531892634875261669235881199, −5.37008119319789691132256949802, −5.21054607338256129646492429297, −4.53555289415880912468798460296, −4.45731172314289042841743476570, −4.42964052663799843463428051341, −4.09677944643574752211361608360, −3.99793999974100759427451722994, −3.51010709223943012825312797950, −3.18134480687261045109326361615, −2.99981978123335405951308899064, −2.71305322515072008659668448732, −2.68156329337505444045101975355, −2.06498841701380841888108266480, −2.00782911250493395679079592510, −1.47445013071255324195551202670, −1.25195164224563536536488446481, −0.877705328346766892143239469982, −0.67518960414272005995134553475, −0.06517781382214689101433544798,
0.06517781382214689101433544798, 0.67518960414272005995134553475, 0.877705328346766892143239469982, 1.25195164224563536536488446481, 1.47445013071255324195551202670, 2.00782911250493395679079592510, 2.06498841701380841888108266480, 2.68156329337505444045101975355, 2.71305322515072008659668448732, 2.99981978123335405951308899064, 3.18134480687261045109326361615, 3.51010709223943012825312797950, 3.99793999974100759427451722994, 4.09677944643574752211361608360, 4.42964052663799843463428051341, 4.45731172314289042841743476570, 4.53555289415880912468798460296, 5.21054607338256129646492429297, 5.37008119319789691132256949802, 5.55531892634875261669235881199, 5.65507343447641795673971559459, 5.81770935679324149884641554258, 6.05287804043078955779494127528, 6.61893424048649174697038790167, 6.73422977769241169477222487143