L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9983621939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9983621939\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
good | 7 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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
−7.84232125263057793666246361886, −7.66981501816775883330459380997, −7.43258593177356800160634817136, −7.14254561741020520523290292026, −6.97670638188966200650886100836, −6.86649025843642921970021235882, −6.16801099675516020279824416309, −6.02856331054450554983654699819, −6.01834080393432966584567233959, −5.89463443287140223083459360678, −5.69430914124502981247248948137, −5.07288703322210171428009826633, −4.75093527236746108051610999977, −4.73415586335395170227535938226, −4.54041723275158527535589520431, −4.46165948994787499370132581884, −3.67476107723092059868908245222, −3.43394796637693852198834991532, −3.10520709311688864591500395430, −3.03312472714709196783661963924, −2.70509528835814710740052894529, −2.62389902806177915964267520957, −2.15081326373838919591372847816, −1.82616607877136577685655652354, −0.890317674643063229284811383072,
0.890317674643063229284811383072, 1.82616607877136577685655652354, 2.15081326373838919591372847816, 2.62389902806177915964267520957, 2.70509528835814710740052894529, 3.03312472714709196783661963924, 3.10520709311688864591500395430, 3.43394796637693852198834991532, 3.67476107723092059868908245222, 4.46165948994787499370132581884, 4.54041723275158527535589520431, 4.73415586335395170227535938226, 4.75093527236746108051610999977, 5.07288703322210171428009826633, 5.69430914124502981247248948137, 5.89463443287140223083459360678, 6.01834080393432966584567233959, 6.02856331054450554983654699819, 6.16801099675516020279824416309, 6.86649025843642921970021235882, 6.97670638188966200650886100836, 7.14254561741020520523290292026, 7.43258593177356800160634817136, 7.66981501816775883330459380997, 7.84232125263057793666246361886