| L(s) = 1 | + 2·9-s + 2·16-s + 4·49-s + 81-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 2·9-s + 2·16-s + 4·49-s + 81-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.050830850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050830850\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( ( 1 + T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 11 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 17 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 19 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 23 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 29 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 41 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{8} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 71 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{2} )^{8} \) |
| 79 | \( ( 1 + T^{2} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
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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
−3.86536533609379565476505078266, −3.82713682089571825057722774892, −3.81637354649738552614106997678, −3.70450563684293059445511879470, −3.66064757833078828388038251897, −3.34878560399564924089880053200, −3.19285214968206362703310353980, −3.09154199417073983588961999013, −2.94617963704626211597248439706, −2.92464980052448675885174946151, −2.74172370240954552767481300389, −2.42916149509417512235505338156, −2.39863028226291123068114558347, −2.31877635417026067321426171168, −2.23254424608303381924714318389, −2.19498383093580978883929385298, −2.01297188200985153874604217827, −1.60686675079266667418326469264, −1.40734440720225911588405286562, −1.39861169822919181815837774998, −1.29444201642253026137785762304, −1.12437406733284238882173013038, −1.08151229521078254367268443930, −0.926820105686623002749626437387, −0.39087849232248869180526932912,
0.39087849232248869180526932912, 0.926820105686623002749626437387, 1.08151229521078254367268443930, 1.12437406733284238882173013038, 1.29444201642253026137785762304, 1.39861169822919181815837774998, 1.40734440720225911588405286562, 1.60686675079266667418326469264, 2.01297188200985153874604217827, 2.19498383093580978883929385298, 2.23254424608303381924714318389, 2.31877635417026067321426171168, 2.39863028226291123068114558347, 2.42916149509417512235505338156, 2.74172370240954552767481300389, 2.92464980052448675885174946151, 2.94617963704626211597248439706, 3.09154199417073983588961999013, 3.19285214968206362703310353980, 3.34878560399564924089880053200, 3.66064757833078828388038251897, 3.70450563684293059445511879470, 3.81637354649738552614106997678, 3.82713682089571825057722774892, 3.86536533609379565476505078266
Plot not available for L-functions of degree greater than 10.
