| L(s) = 1 | + 4·9-s − 8·25-s − 16·37-s + 16·43-s + 16·67-s + 48·79-s + 2·81-s + 64·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 8/5·25-s − 2.63·37-s + 2.43·43-s + 1.95·67-s + 5.40·79-s + 2/9·81-s + 6.13·109-s + 4.36·121-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 + 4.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(25.11951101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.11951101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 4 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 314 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T^{2} + 420 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 12 T^{2} + 246 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 16 T^{2} + 1050 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 32 T^{2} + 1546 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 20 T^{2} + 1630 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 100 T^{2} + 5524 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 172 T^{2} + 11806 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 108 T^{2} + 5942 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 108 T^{2} + 6678 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 224 T^{2} + 19984 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 176 T^{2} + 14938 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 240 T^{2} + 24480 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 300 T^{2} + 36078 T^{4} + 300 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 156 T^{2} + 20676 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 48 T^{2} + 17472 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.78166466599217414043756423997, −3.55945285947619058407763342777, −3.49265946101668150598600388562, −3.36598525640131465419371797487, −3.30590385948505265437173368800, −3.14863640194506680196917307027, −3.09621031944546563618237944223, −2.96526903935628234833896452797, −2.88324087098089827778870476841, −2.74142426589231869464274810510, −2.41228966472753479693467048353, −2.13344706017708803155942038525, −2.04838518543682855371059336965, −1.97531050582278187542881742510, −1.93622340817762792470535581922, −1.84479442914382771022761530873, −1.83884939316778028211719846334, −1.83172391767441304572286680314, −1.51327049410459773077120050206, −0.902826122684584530674902266770, −0.77593416254338411940302833134, −0.76502208643221680785449216490, −0.76006218763835575960900461857, −0.56166021344069638257527575810, −0.40243226351234913036154111119,
0.40243226351234913036154111119, 0.56166021344069638257527575810, 0.76006218763835575960900461857, 0.76502208643221680785449216490, 0.77593416254338411940302833134, 0.902826122684584530674902266770, 1.51327049410459773077120050206, 1.83172391767441304572286680314, 1.83884939316778028211719846334, 1.84479442914382771022761530873, 1.93622340817762792470535581922, 1.97531050582278187542881742510, 2.04838518543682855371059336965, 2.13344706017708803155942038525, 2.41228966472753479693467048353, 2.74142426589231869464274810510, 2.88324087098089827778870476841, 2.96526903935628234833896452797, 3.09621031944546563618237944223, 3.14863640194506680196917307027, 3.30590385948505265437173368800, 3.36598525640131465419371797487, 3.49265946101668150598600388562, 3.55945285947619058407763342777, 3.78166466599217414043756423997
Plot not available for L-functions of degree greater than 10.
