| L(s) = 1 | + 8·2-s + 36·4-s + 120·8-s + 330·16-s + 16·25-s + 792·32-s + 64·43-s − 4·49-s + 128·50-s + 1.71e3·64-s − 32·71-s − 48·79-s + 10·81-s + 512·86-s − 32·98-s + 576·100-s − 88·121-s + 127-s + 3.43e3·128-s + 131-s + 137-s + 139-s − 256·142-s + 149-s + 151-s + 157-s − 384·158-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s + 42.4·8-s + 82.5·16-s + 16/5·25-s + 140.·32-s + 9.75·43-s − 4/7·49-s + 18.1·50-s + 214.5·64-s − 3.79·71-s − 5.40·79-s + 10/9·81-s + 55.2·86-s − 3.23·98-s + 57.5·100-s − 8·121-s + 0.0887·127-s + 303.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 21.4·142-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 30.5·158-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(418.1499070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(418.1499070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 - 10 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 + 4 T^{2} - 10 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | \( 1 + 310 T^{4} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 - 8 T^{2} + 38 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 12 T^{2} + 166 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 24 T^{2} + 838 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 28 T^{2} + 246 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 84 T^{2} + 3334 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 84 T^{2} + 3574 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 116 T^{2} + 5990 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 52 T^{2} + 3926 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 108 T^{2} + 6886 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 148 T^{2} + 10086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 216 T^{2} + 18598 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 192 T^{2} + 16630 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 92 T^{2} + 3926 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 124 T^{2} + 9014 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 152 T^{2} + 18182 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 188 T^{2} + 19190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 172 T^{2} + 26102 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.58696474229942634945946628329, −4.45865238423089957367609717624, −4.41763164738188979053493111006, −4.26993994497284867154739850293, −4.24145408803672970481761681498, −4.18957740480635236303461343863, −3.99643544585557338642938649104, −3.95464164930227558061773156770, −3.83190489830823968882272076281, −3.32161872110317103064426462375, −3.31159540622794470134586252896, −3.25120527139660967557555024280, −2.98843241552799605161279797847, −2.82438440641602750785963306709, −2.80658277856213348852764073029, −2.80302607631594255486200183643, −2.39596921077032745336579671841, −2.29242286878425419671947571755, −2.29183839840830192334422019742, −2.19803920519537568686586169551, −1.54778598981722576761583854287, −1.39773180230670617994151553922, −1.26678522265415791953678982884, −1.11320497805566594274401066309, −0.76832849162960674276123471425,
0.76832849162960674276123471425, 1.11320497805566594274401066309, 1.26678522265415791953678982884, 1.39773180230670617994151553922, 1.54778598981722576761583854287, 2.19803920519537568686586169551, 2.29183839840830192334422019742, 2.29242286878425419671947571755, 2.39596921077032745336579671841, 2.80302607631594255486200183643, 2.80658277856213348852764073029, 2.82438440641602750785963306709, 2.98843241552799605161279797847, 3.25120527139660967557555024280, 3.31159540622794470134586252896, 3.32161872110317103064426462375, 3.83190489830823968882272076281, 3.95464164930227558061773156770, 3.99643544585557338642938649104, 4.18957740480635236303461343863, 4.24145408803672970481761681498, 4.26993994497284867154739850293, 4.41763164738188979053493111006, 4.45865238423089957367609717624, 4.58696474229942634945946628329
Plot not available for L-functions of degree greater than 10.
