| L(s) = 1 | − 4·19-s − 4·41-s − 8·59-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 4·19-s − 4·41-s − 8·59-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3393751014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3393751014\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{8} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71156282822601169548260586302, −6.61808746312110430846732915380, −6.45335709132326780459928504062, −6.37475380718178437349379253851, −6.08920768080841856083918899542, −6.07622301929319422954706774096, −5.52988132017086683409340862898, −5.41196162029333550606420666417, −5.28394355768444367825233753659, −4.73436185053315994444534527764, −4.56025821499251120501190033874, −4.54605955581184420751376786667, −4.38999806617386669448990219108, −4.24248952825084641288886257553, −3.63717486889119172252718859434, −3.52831827802221885680854809841, −3.17141705400259499739940157965, −3.12359012735038097370924039648, −2.84813675121973321726684477036, −2.35064442244631507813249404902, −1.91812195719721418456131785844, −1.89992714791620668640772235271, −1.71581791989755751048244052630, −1.32656690544536693158679870940, −0.31114918539938107097133791348,
0.31114918539938107097133791348, 1.32656690544536693158679870940, 1.71581791989755751048244052630, 1.89992714791620668640772235271, 1.91812195719721418456131785844, 2.35064442244631507813249404902, 2.84813675121973321726684477036, 3.12359012735038097370924039648, 3.17141705400259499739940157965, 3.52831827802221885680854809841, 3.63717486889119172252718859434, 4.24248952825084641288886257553, 4.38999806617386669448990219108, 4.54605955581184420751376786667, 4.56025821499251120501190033874, 4.73436185053315994444534527764, 5.28394355768444367825233753659, 5.41196162029333550606420666417, 5.52988132017086683409340862898, 6.07622301929319422954706774096, 6.08920768080841856083918899542, 6.37475380718178437349379253851, 6.45335709132326780459928504062, 6.61808746312110430846732915380, 6.71156282822601169548260586302
