L(s) = 1 | + 2·7-s − 2·11-s + 4·17-s − 2·23-s + 25-s − 2·31-s − 2·41-s + 2·49-s − 4·53-s + 2·67-s − 4·71-s + 4·73-s − 4·77-s + 2·101-s − 2·103-s + 4·107-s + 2·113-s + 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯ |
L(s) = 1 | + 2·7-s − 2·11-s + 4·17-s − 2·23-s + 25-s − 2·31-s − 2·41-s + 2·49-s − 4·53-s + 2·67-s − 4·71-s + 4·73-s − 4·77-s + 2·101-s − 2·103-s + 4·107-s + 2·113-s + 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.305430952\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.305430952\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−7.13790900736843269939375503332, −6.56651311369008780948819692533, −6.30795519151329098363724291381, −6.27849522966770459385418135438, −6.05478299350966692998965031741, −5.66744758263255317793636553171, −5.54546492928011380165718243405, −5.33402855952051040018954262106, −5.05587261847517322266761732452, −5.03523700319793670395609716822, −4.94785486995268190644900432953, −4.61489944507902507613442199574, −4.40052075700658960799858091049, −3.85248486766315455387593720460, −3.60193394488654963506184967820, −3.48395694502500866598262326846, −3.45995092856768821334485277478, −3.06355604173032085187074990485, −2.57351230167932303089028828760, −2.53009052315635872044885688369, −2.11681672933630524361109102582, −1.57088094904265804586296582513, −1.50796677308599097424015984281, −1.50602115763110178260499011098, −0.65225055960135622747187196493,
0.65225055960135622747187196493, 1.50602115763110178260499011098, 1.50796677308599097424015984281, 1.57088094904265804586296582513, 2.11681672933630524361109102582, 2.53009052315635872044885688369, 2.57351230167932303089028828760, 3.06355604173032085187074990485, 3.45995092856768821334485277478, 3.48395694502500866598262326846, 3.60193394488654963506184967820, 3.85248486766315455387593720460, 4.40052075700658960799858091049, 4.61489944507902507613442199574, 4.94785486995268190644900432953, 5.03523700319793670395609716822, 5.05587261847517322266761732452, 5.33402855952051040018954262106, 5.54546492928011380165718243405, 5.66744758263255317793636553171, 6.05478299350966692998965031741, 6.27849522966770459385418135438, 6.30795519151329098363724291381, 6.56651311369008780948819692533, 7.13790900736843269939375503332