L(s) = 1 | + 4-s + 5-s + 16-s + 20-s + 8·23-s − 31-s + 2·37-s + 47-s + 49-s − 2·53-s + 59-s + 4·67-s − 2·71-s + 80-s − 16·89-s + 8·92-s − 97-s − 103-s + 113-s + 8·115-s − 124-s + 125-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + ⋯ |
L(s) = 1 | + 4-s + 5-s + 16-s + 20-s + 8·23-s − 31-s + 2·37-s + 47-s + 49-s − 2·53-s + 59-s + 4·67-s − 2·71-s + 80-s − 16·89-s + 8·92-s − 97-s − 103-s + 113-s + 8·115-s − 124-s + 125-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.559893368\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.559893368\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 7 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( ( 1 - T + T^{2} )^{8} \) |
| 29 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 43 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 53 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 67 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 71 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \) |
| 89 | \( ( 1 + T )^{16} \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.01939688444183918602044045407, −3.45635155482939134579688994336, −3.43077939147794961493488788904, −3.36781643969641202045126232185, −3.12147147766126177954214330555, −3.09851753439624664637011447344, −3.02362605317537372332403189386, −2.97825849042611852837872480355, −2.94747805296358141152182944609, −2.88127019819479572893797242291, −2.62581748094913698932389524942, −2.44720772116325749108722256702, −2.40462589015510938630845445862, −2.26549419501268866170171849637, −2.18730030709731793448207395987, −2.15532165277758020528918703446, −1.67144306035248999820036698691, −1.45490200566881147822610228346, −1.41434960442159240740548448588, −1.38893199992097121535414708850, −1.26888333004811860866723755382, −1.26441198886815818114745929824, −0.994947766909477974499249911054, −0.794313357271090733583329391944, −0.49757716454336614095278597182,
0.49757716454336614095278597182, 0.794313357271090733583329391944, 0.994947766909477974499249911054, 1.26441198886815818114745929824, 1.26888333004811860866723755382, 1.38893199992097121535414708850, 1.41434960442159240740548448588, 1.45490200566881147822610228346, 1.67144306035248999820036698691, 2.15532165277758020528918703446, 2.18730030709731793448207395987, 2.26549419501268866170171849637, 2.40462589015510938630845445862, 2.44720772116325749108722256702, 2.62581748094913698932389524942, 2.88127019819479572893797242291, 2.94747805296358141152182944609, 2.97825849042611852837872480355, 3.02362605317537372332403189386, 3.09851753439624664637011447344, 3.12147147766126177954214330555, 3.36781643969641202045126232185, 3.43077939147794961493488788904, 3.45635155482939134579688994336, 4.01939688444183918602044045407
Plot not available for L-functions of degree greater than 10.