| L(s) = 1 | − 16·9-s + 32·25-s + 20·49-s + 124·81-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 512·225-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 5.33·9-s + 32/5·25-s + 20/7·49-s + 13.7·81-s + 3.63·121-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 + 2.46·169-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 + 0.0669·223-s − 34.1·225-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.502937371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.502937371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76285928726113166793751585330, −3.63680198643070473208835046390, −3.55488630590315406052035984153, −3.35495826959218313069954254676, −3.28417549308225109353295217582, −3.14961474836494365467544485114, −3.09752441848360640054028096889, −3.09311379520364542955080406735, −3.03915795818782776745798006816, −2.78724265167491266427991605514, −2.60221142640196435238869658943, −2.58496958826179884167791594837, −2.52646499416516191181961122249, −2.25470791701218558743337775310, −2.07598010670868475954914178983, −2.03092907925737576382908875225, −1.99416831950171270366876798606, −1.53899649131809756130675819413, −1.38450297121084948914029197418, −1.14373472265788784574035500827, −0.854964448119314014011735169621, −0.839749707816597579851495690562, −0.54931187676115688428746789701, −0.46051135404297379575111914218, −0.37564449875954258506866882141,
0.37564449875954258506866882141, 0.46051135404297379575111914218, 0.54931187676115688428746789701, 0.839749707816597579851495690562, 0.854964448119314014011735169621, 1.14373472265788784574035500827, 1.38450297121084948914029197418, 1.53899649131809756130675819413, 1.99416831950171270366876798606, 2.03092907925737576382908875225, 2.07598010670868475954914178983, 2.25470791701218558743337775310, 2.52646499416516191181961122249, 2.58496958826179884167791594837, 2.60221142640196435238869658943, 2.78724265167491266427991605514, 3.03915795818782776745798006816, 3.09311379520364542955080406735, 3.09752441848360640054028096889, 3.14961474836494365467544485114, 3.28417549308225109353295217582, 3.35495826959218313069954254676, 3.55488630590315406052035984153, 3.63680198643070473208835046390, 3.76285928726113166793751585330
Plot not available for L-functions of degree greater than 10.
