| L(s) = 1 | − 4·4-s + 12·16-s − 8·25-s + 32·37-s + 14·49-s − 32·64-s + 32·100-s − 40·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 56·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 8/5·25-s + 5.26·37-s + 2·49-s − 4·64-s + 16/5·100-s − 3.83·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9666661695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9666661695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.935129393608426394090796395116, −8.425704348451493873072838903100, −8.161502421217799917727649709974, −8.085618846099869430139375094845, −8.044363781605078325827204284040, −7.37665862797470686348137423925, −7.37165960470717459832508723330, −7.24233074393936968529389918812, −6.41829084372774237450978754295, −6.30568558233015063402682851566, −5.99047525727115302996760780973, −5.83225216106266373223867247245, −5.49932079956869419694866465152, −5.20088030260710437746340211427, −4.90968114685182412563974701048, −4.41873339520697452937231567731, −4.19471198274640899982973405615, −4.05001697945268262052882811754, −3.96560513896203749463257506335, −3.18078334605983012449752342670, −2.98047910348469955276902365736, −2.49641330283895631484505564332, −1.97703008993395072583230577514, −1.12952791476287571249147936987, −0.65103277755109469888434596268,
0.65103277755109469888434596268, 1.12952791476287571249147936987, 1.97703008993395072583230577514, 2.49641330283895631484505564332, 2.98047910348469955276902365736, 3.18078334605983012449752342670, 3.96560513896203749463257506335, 4.05001697945268262052882811754, 4.19471198274640899982973405615, 4.41873339520697452937231567731, 4.90968114685182412563974701048, 5.20088030260710437746340211427, 5.49932079956869419694866465152, 5.83225216106266373223867247245, 5.99047525727115302996760780973, 6.30568558233015063402682851566, 6.41829084372774237450978754295, 7.24233074393936968529389918812, 7.37165960470717459832508723330, 7.37665862797470686348137423925, 8.044363781605078325827204284040, 8.085618846099869430139375094845, 8.161502421217799917727649709974, 8.425704348451493873072838903100, 8.935129393608426394090796395116
