| L(s) = 1 | − 4·7-s + 16-s + 4·31-s + 10·49-s − 4·79-s − 4·97-s − 4·112-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 16·217-s + ⋯ |
| L(s) = 1 | − 4·7-s + 16-s + 4·31-s + 10·49-s − 4·79-s − 4·97-s − 4·112-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 16·217-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7943653069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7943653069\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 7 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 13 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 41 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{8} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 67 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 + T^{2} )^{8} \) |
| 79 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 83 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
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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
−4.11117974003182154835377785348, −4.10138376469841561499873327468, −3.94916829315224528102807916422, −3.55859354667974660930774562798, −3.52452093217410266029486365937, −3.39402786809487850388162923236, −3.22985522310687576205902997524, −3.16899701042474434620135318813, −3.14676131367927703229907701629, −3.02660466131275230003323295219, −2.91248591376626704475575134112, −2.76786616850989997190009254096, −2.74075989518355010926395928147, −2.49221584353409145990665859967, −2.42491946892223110330033464546, −2.30091691347588048489518700279, −1.92480651337774270848425548859, −1.86913406304135666342827114457, −1.76723085765776005991735677568, −1.58271735753355434006350704839, −1.26585062887823007693960992745, −1.00783614577151169499100071847, −0.882020100585218365574229804292, −0.68716964152145887877836181788, −0.51777592074936686656825626001,
0.51777592074936686656825626001, 0.68716964152145887877836181788, 0.882020100585218365574229804292, 1.00783614577151169499100071847, 1.26585062887823007693960992745, 1.58271735753355434006350704839, 1.76723085765776005991735677568, 1.86913406304135666342827114457, 1.92480651337774270848425548859, 2.30091691347588048489518700279, 2.42491946892223110330033464546, 2.49221584353409145990665859967, 2.74075989518355010926395928147, 2.76786616850989997190009254096, 2.91248591376626704475575134112, 3.02660466131275230003323295219, 3.14676131367927703229907701629, 3.16899701042474434620135318813, 3.22985522310687576205902997524, 3.39402786809487850388162923236, 3.52452093217410266029486365937, 3.55859354667974660930774562798, 3.94916829315224528102807916422, 4.10138376469841561499873327468, 4.11117974003182154835377785348
Plot not available for L-functions of degree greater than 10.
