| L(s) = 1 | + 2·9-s − 8·11-s + 16-s − 4·43-s + 4·59-s + 81-s − 4·83-s + 4·97-s − 16·99-s − 4·107-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s − 8·176-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·9-s − 8·11-s + 16-s − 4·43-s + 4·59-s + 81-s − 4·83-s + 4·97-s − 16·99-s − 4·107-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s − 8·176-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0002413113241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0002413113241\) |
| \(L(1)\) |
|
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 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | \( 1 - T^{4} + T^{8} \) |
| good | 5 | \( 1 - T^{8} + T^{16} \) |
| 7 | \( 1 - T^{8} + T^{16} \) |
| 11 | \( ( 1 + T )^{8}( 1 - T^{4} + T^{8} ) \) |
| 13 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 19 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( 1 - T^{8} + T^{16} \) |
| 29 | \( 1 - T^{8} + T^{16} \) |
| 31 | \( 1 - T^{8} + T^{16} \) |
| 37 | \( ( 1 + T^{8} )^{2} \) |
| 41 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 + T^{4} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 - T + T^{2} )^{4}( 1 + T^{2} )^{4} \) |
| 61 | \( 1 - T^{8} + T^{16} \) |
| 67 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( 1 - T^{8} + T^{16} \) |
| 83 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 - T + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
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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
−4.47991957180521977566784910372, −4.37687917740981633462104463911, −4.32685756189131148015430243577, −4.02367012838623944818791291820, −3.72241139277576827031375825266, −3.69955911028254979234470266749, −3.53973172145381216966427702709, −3.41551467322052247982345986845, −3.33843568530003383231623374809, −3.14464570852643227162856587452, −3.11580570595110078650162082915, −2.88196176739983527426377468143, −2.84715385184354592229401556803, −2.54443935499194259395267591006, −2.48953842715449935363919793720, −2.25934851205809238318907065960, −2.25789506771108641155942810402, −2.21094680216179537342340585705, −1.98450103875216767823258243688, −1.91713421047311612487806105333, −1.48096616127344466775523296868, −1.24040479045391677000560704141, −1.07396773498078002113915333332, −0.999077228290856627203969613894, −0.008838924420423837138712630765,
0.008838924420423837138712630765, 0.999077228290856627203969613894, 1.07396773498078002113915333332, 1.24040479045391677000560704141, 1.48096616127344466775523296868, 1.91713421047311612487806105333, 1.98450103875216767823258243688, 2.21094680216179537342340585705, 2.25789506771108641155942810402, 2.25934851205809238318907065960, 2.48953842715449935363919793720, 2.54443935499194259395267591006, 2.84715385184354592229401556803, 2.88196176739983527426377468143, 3.11580570595110078650162082915, 3.14464570852643227162856587452, 3.33843568530003383231623374809, 3.41551467322052247982345986845, 3.53973172145381216966427702709, 3.69955911028254979234470266749, 3.72241139277576827031375825266, 4.02367012838623944818791291820, 4.32685756189131148015430243577, 4.37687917740981633462104463911, 4.47991957180521977566784910372
Plot not available for L-functions of degree greater than 10.
