L(s) = 1 | + 2·16-s + 4·25-s + 2·49-s − 8·67-s − 20·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + ⋯ |
L(s) = 1 | + 2·16-s + 4·25-s + 2·49-s − 8·67-s − 20·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2270370918\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2270370918\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 11 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
good | 2 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 5 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 13 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 19 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 23 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 29 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 37 | \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{8} \) |
| 71 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 79 | \( ( 1 + T )^{16}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{2} )^{16} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.96516354527302468125286854457, −2.95567458311936212579182386401, −2.93546457510629599122432856361, −2.82611081045605333172500756675, −2.79757146720682771443586220879, −2.79345917471770236625237965966, −2.74640021440156624103945150881, −2.63805802925140126106760694773, −2.47461377835267084633176270288, −2.46170051947185904185253277059, −2.35405747009839158525114193277, −2.27602869291818794927641599057, −2.07150309078283174174464477932, −1.89201622697530961267569647771, −1.71686687436873811534454076603, −1.59401119483080453256658171939, −1.57709066657913199015786031238, −1.56068324131744474829727750197, −1.53608569331353593124517473024, −1.35212972536604054835604403879, −1.26000879937970853575463934527, −1.23470822574348972553277143840, −1.16817165119937264475905190244, −1.04958964635061490947976977415, −0.63545503605285065932662652634,
0.63545503605285065932662652634, 1.04958964635061490947976977415, 1.16817165119937264475905190244, 1.23470822574348972553277143840, 1.26000879937970853575463934527, 1.35212972536604054835604403879, 1.53608569331353593124517473024, 1.56068324131744474829727750197, 1.57709066657913199015786031238, 1.59401119483080453256658171939, 1.71686687436873811534454076603, 1.89201622697530961267569647771, 2.07150309078283174174464477932, 2.27602869291818794927641599057, 2.35405747009839158525114193277, 2.46170051947185904185253277059, 2.47461377835267084633176270288, 2.63805802925140126106760694773, 2.74640021440156624103945150881, 2.79345917471770236625237965966, 2.79757146720682771443586220879, 2.82611081045605333172500756675, 2.93546457510629599122432856361, 2.95567458311936212579182386401, 2.96516354527302468125286854457
Plot not available for L-functions of degree greater than 10.