| L(s) = 1 | − 16·13-s + 16·37-s + 24·49-s + 48·53-s + 16·61-s + 4·81-s + 48·101-s + 64·109-s + 48·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 2.63·37-s + 24/7·49-s + 6.59·53-s + 2.04·61-s + 4/9·81-s + 4.77·101-s + 6.13·109-s + 4.51·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276150409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276150409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 T^{4} - 26 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 34 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 68 T^{4} - 12762 T^{8} - 68 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 120 T^{3} + 446 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 124 T^{4} - 27546 T^{8} + 124 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 44 T^{2} + 1110 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1022 T^{4} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 312 T^{3} + 3038 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 452 T^{4} - 2831322 T^{8} - 452 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 24 T + 288 T^{2} - 2856 T^{3} + 23966 T^{4} - 2856 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 9668 T^{4} + 43107750 T^{8} - 9668 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 8 T + 32 T^{2} - 504 T^{3} + 7934 T^{4} - 504 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 4732 T^{4} + 41046246 T^{8} + 4732 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 236 T^{2} + 23574 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 60 T^{2} + 1094 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 8188 T^{4} + 45970278 T^{8} + 8188 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 182 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.41175964300818403089669839294, −5.28932573188789348279150091371, −5.07243271396618011248734780876, −4.99173657453981095464944475623, −4.84352508277310901171769690462, −4.75394476549934318326248513629, −4.40742518223259501489448033198, −4.39323780303103347087637906026, −4.23496479043453044143770930483, −4.05869649589361875992237546028, −3.99056227180618458706877151004, −3.55327633893016097727842944189, −3.51149908649555035306620279972, −3.41662618134599008689312620583, −3.08480345280821336825456132144, −2.73109019078727698079115608358, −2.55318684750207342388014435224, −2.52861159596849900746909822271, −2.27455952365988890164315916108, −2.22751409905012043109948860848, −2.04329192091514271459314444020, −1.92363720144388450694252278766, −0.872220906877429281569831727505, −0.812210655585464839053756116890, −0.70366005938028500167467444847,
0.70366005938028500167467444847, 0.812210655585464839053756116890, 0.872220906877429281569831727505, 1.92363720144388450694252278766, 2.04329192091514271459314444020, 2.22751409905012043109948860848, 2.27455952365988890164315916108, 2.52861159596849900746909822271, 2.55318684750207342388014435224, 2.73109019078727698079115608358, 3.08480345280821336825456132144, 3.41662618134599008689312620583, 3.51149908649555035306620279972, 3.55327633893016097727842944189, 3.99056227180618458706877151004, 4.05869649589361875992237546028, 4.23496479043453044143770930483, 4.39323780303103347087637906026, 4.40742518223259501489448033198, 4.75394476549934318326248513629, 4.84352508277310901171769690462, 4.99173657453981095464944475623, 5.07243271396618011248734780876, 5.28932573188789348279150091371, 5.41175964300818403089669839294
Plot not available for L-functions of degree greater than 10.
