L(s) = 1 | + 2·7-s + 16-s + 2·19-s − 2·31-s − 2·37-s + 6·43-s + 49-s − 4·67-s − 4·73-s − 2·97-s − 6·103-s + 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 2·7-s + 16-s + 2·19-s − 2·31-s − 2·37-s + 6·43-s + 49-s − 4·67-s − 4·73-s − 2·97-s − 6·103-s + 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.131654088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131654088\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.57247009896421151528779520251, −7.43447111586758402498875781759, −7.25704756397129989719478031650, −7.00413122506102861104146279792, −6.95155218175250706796369691965, −6.12184918493857249235578019908, −6.08236340971301555016360185405, −5.88050261215556366916036047167, −5.85227941283572077116123002544, −5.36327450569760800441968400123, −5.14099150499034411808459670238, −5.04829324274203416652598710843, −5.02020567025159185866521503885, −4.28500326620099445323270361678, −4.11377982116627186934721578177, −4.04986296281894392663774182877, −3.93846544203856168203041988128, −3.31052947040954895929487667957, −2.94263895845517460930360134371, −2.80366753766539992535523215536, −2.62153557837044334587783572346, −2.02337167280871345113185974527, −1.45823378368166458461027255327, −1.39005318553599220502305882319, −1.27520525454857803587190793091,
1.27520525454857803587190793091, 1.39005318553599220502305882319, 1.45823378368166458461027255327, 2.02337167280871345113185974527, 2.62153557837044334587783572346, 2.80366753766539992535523215536, 2.94263895845517460930360134371, 3.31052947040954895929487667957, 3.93846544203856168203041988128, 4.04986296281894392663774182877, 4.11377982116627186934721578177, 4.28500326620099445323270361678, 5.02020567025159185866521503885, 5.04829324274203416652598710843, 5.14099150499034411808459670238, 5.36327450569760800441968400123, 5.85227941283572077116123002544, 5.88050261215556366916036047167, 6.08236340971301555016360185405, 6.12184918493857249235578019908, 6.95155218175250706796369691965, 7.00413122506102861104146279792, 7.25704756397129989719478031650, 7.43447111586758402498875781759, 7.57247009896421151528779520251