| L(s) = 1 | − 7-s − 2·13-s − 4·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s + 61-s − 67-s − 2·73-s + 2·79-s + 2·91-s − 2·97-s − 103-s − 2·109-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 7-s − 2·13-s − 4·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s + 61-s − 67-s − 2·73-s + 2·79-s + 2·91-s − 2·97-s − 103-s − 2·109-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01814828836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01814828836\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.886808015529885984808044328872, −8.402467183063853393728792728529, −8.252037828279487126816364464705, −7.74440841320050397953276222343, −7.22939508447414411011579154741, −6.88121592284180718574654588969, −6.80933246713742258843350578098, −6.26321941633441879256100865509, −6.00489340793297580582579977728, −5.45078450852989825533104866316, −5.12933924509255600012966557161, −4.62913795408109624020504261532, −4.23420722478004079727463646490, −3.88429480254012451373613415065, −3.54376519921051401387797530369, −2.74842833700036690538867205464, −2.53149522541700497962983290257, −1.94309415356894074357782550771, −1.73211871609483328761125436127, −0.06820835859080096726495461965,
0.06820835859080096726495461965, 1.73211871609483328761125436127, 1.94309415356894074357782550771, 2.53149522541700497962983290257, 2.74842833700036690538867205464, 3.54376519921051401387797530369, 3.88429480254012451373613415065, 4.23420722478004079727463646490, 4.62913795408109624020504261532, 5.12933924509255600012966557161, 5.45078450852989825533104866316, 6.00489340793297580582579977728, 6.26321941633441879256100865509, 6.80933246713742258843350578098, 6.88121592284180718574654588969, 7.22939508447414411011579154741, 7.74440841320050397953276222343, 8.252037828279487126816364464705, 8.402467183063853393728792728529, 8.886808015529885984808044328872
