| L(s) = 1 | − 8·7-s + 16·13-s + 4·16-s + 16·31-s + 16·37-s − 8·43-s + 32·49-s − 8·61-s + 40·67-s − 8·73-s − 128·91-s − 32·97-s − 8·103-s − 32·112-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 4.43·13-s + 16-s + 2.87·31-s + 2.63·37-s − 1.21·43-s + 32/7·49-s − 1.02·61-s + 4.88·67-s − 0.936·73-s − 13.4·91-s − 3.24·97-s − 0.788·103-s − 3.02·112-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047501397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047501397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 4 T + 8 T^{2} + 24 T^{3} + 71 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 28 T^{2} + 384 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 144 T^{3} + 623 T^{4} - 144 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 76 T^{4} - 111450 T^{8} - 76 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 1424 T^{4} + 1024290 T^{8} + 1424 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 20 T^{2} + 1728 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 144 T^{3} + 287 T^{4} - 144 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 28 T^{2} - 816 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T + 8 T^{2} + 132 T^{3} + 2078 T^{4} + 132 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 3964 T^{4} + 8158086 T^{8} - 3964 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 8464 T^{4} + 31941186 T^{8} - 8464 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 4 T^{2} + 1566 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 20 T + 200 T^{2} - 2280 T^{3} + 23783 T^{4} - 2280 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 100 T^{2} + 12528 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 8 T^{2} + 192 T^{3} + 3983 T^{4} + 192 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 286 T^{2} + 32715 T^{4} - 286 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 17456 T^{4} + 170554626 T^{8} + 17456 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 212 T^{2} + 25134 T^{4} + 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 864 T^{3} + 3983 T^{4} + 864 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.67014416964302427701792958459, −4.27593671686820238937476843026, −4.13409198033614155900197758655, −4.00796006719401609515718863369, −3.87937916309513500933153237229, −3.78720018548788281523483844961, −3.75265136138909991804666987155, −3.63769503207605288516546458393, −3.54152613561340717493771787183, −3.46626303997742316842837949032, −3.00025432505431374359389667766, −2.92175924988487789928649943215, −2.89802433601626935590418911500, −2.86427287380545648201944976475, −2.60967387060354627877961696769, −2.46582085224903744607586868241, −2.30569340193466962867258513024, −1.96465373250865687154178706526, −1.72131462526024312967685766431, −1.39571908671945850645024373831, −1.27079658793779599653768534130, −1.08104442099818857518593651876, −0.914050157875141124082925466843, −0.815838713256878918002167900146, −0.21055746666800157000662563466,
0.21055746666800157000662563466, 0.815838713256878918002167900146, 0.914050157875141124082925466843, 1.08104442099818857518593651876, 1.27079658793779599653768534130, 1.39571908671945850645024373831, 1.72131462526024312967685766431, 1.96465373250865687154178706526, 2.30569340193466962867258513024, 2.46582085224903744607586868241, 2.60967387060354627877961696769, 2.86427287380545648201944976475, 2.89802433601626935590418911500, 2.92175924988487789928649943215, 3.00025432505431374359389667766, 3.46626303997742316842837949032, 3.54152613561340717493771787183, 3.63769503207605288516546458393, 3.75265136138909991804666987155, 3.78720018548788281523483844961, 3.87937916309513500933153237229, 4.00796006719401609515718863369, 4.13409198033614155900197758655, 4.27593671686820238937476843026, 4.67014416964302427701792958459
Plot not available for L-functions of degree greater than 10.
