| L(s) = 1 | + 8·9-s + 16·25-s + 32·41-s − 24·49-s − 16·73-s + 20·81-s − 16·89-s + 64·97-s − 16·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 16/5·25-s + 4.99·41-s − 3.42·49-s − 1.87·73-s + 20/9·81-s − 1.69·89-s + 6.49·97-s − 1.50·113-s + 8/11·121-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 + 6.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.530215340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.530215340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T^{2} + 238 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 40 T^{2} + 706 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 36 T^{2} + 654 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 40 T^{2} + 1794 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 40 T^{2} + 2338 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 164 T^{2} + 10414 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 136 T^{2} + 9954 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 228 T^{2} + 19950 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 72 T^{2} + 1538 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 164 T^{2} + 13390 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 132 T^{2} + 10446 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.39054769868073767882556639139, −4.32885288161913632559807493785, −3.91966559040559718073868123322, −3.78353067690999799412396844091, −3.75797545982998771348245032454, −3.59257996321499446884508658815, −3.50370917695426582723481218536, −3.49972059029058729477312004777, −3.19595605589059963400216967706, −2.90336035081715061912122984349, −2.87006109881376276704571621691, −2.82400656246198114403080816276, −2.63667081987005920878639050349, −2.56789771097012611614956634060, −2.28332866909072352122555587152, −2.01404384087698047113976413446, −1.92993837891286044303815622527, −1.85878700800801267259034683495, −1.55132880330395689650791014964, −1.45758768056496823398091512716, −1.08456016790643178282942107262, −1.05921171224173265965769396565, −0.877668712304140434119579033523, −0.77605242185660220359820470352, −0.23488572646248179933581583897,
0.23488572646248179933581583897, 0.77605242185660220359820470352, 0.877668712304140434119579033523, 1.05921171224173265965769396565, 1.08456016790643178282942107262, 1.45758768056496823398091512716, 1.55132880330395689650791014964, 1.85878700800801267259034683495, 1.92993837891286044303815622527, 2.01404384087698047113976413446, 2.28332866909072352122555587152, 2.56789771097012611614956634060, 2.63667081987005920878639050349, 2.82400656246198114403080816276, 2.87006109881376276704571621691, 2.90336035081715061912122984349, 3.19595605589059963400216967706, 3.49972059029058729477312004777, 3.50370917695426582723481218536, 3.59257996321499446884508658815, 3.75797545982998771348245032454, 3.78353067690999799412396844091, 3.91966559040559718073868123322, 4.32885288161913632559807493785, 4.39054769868073767882556639139
Plot not available for L-functions of degree greater than 10.
