| L(s) = 1 | + 2·2-s + 4·3-s + 2·4-s + 8·6-s + 8·9-s − 8·11-s + 8·12-s + 8·17-s + 16·18-s − 16·22-s + 12·27-s + 8·32-s − 32·33-s + 16·34-s + 16·36-s − 8·41-s − 28·43-s − 16·44-s + 32·51-s + 24·54-s + 16·64-s − 64·66-s + 28·67-s + 16·68-s − 16·73-s + 24·81-s − 16·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 4-s + 3.26·6-s + 8/3·9-s − 2.41·11-s + 2.30·12-s + 1.94·17-s + 3.77·18-s − 3.41·22-s + 2.30·27-s + 1.41·32-s − 5.57·33-s + 2.74·34-s + 8/3·36-s − 1.24·41-s − 4.26·43-s − 2.41·44-s + 4.48·51-s + 3.26·54-s + 2·64-s − 7.87·66-s + 3.42·67-s + 1.94·68-s − 1.87·73-s + 8/3·81-s − 1.76·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.279745056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.279745056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - 2 T + 2 T^{2} - 2 T^{3} - 2 T^{4} - 2 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 24 T^{4} + 3326 T^{8} - 24 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 4 T^{4} - 24794 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 984 T^{4} + 581246 T^{8} - 984 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 76 T^{2} + 2806 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 24 T^{2} + 1566 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 2876 T^{4} + 4505446 T^{8} + 2876 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 14 T + 98 T^{2} + 910 T^{3} + 7966 T^{4} + 910 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 3544 T^{4} + 6827326 T^{8} - 3544 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 7356 T^{4} + 26359526 T^{8} + 7356 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 164 T^{2} + 12406 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 144 T^{2} + 10206 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 14 T + 98 T^{2} - 1246 T^{3} + 15358 T^{4} - 1246 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 104 T^{2} + 11166 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 32 T^{2} - 72 T^{3} - 6562 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 156 T^{2} + 13446 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 22 T + 242 T^{2} - 3102 T^{3} + 36398 T^{4} - 3102 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 308 T^{2} + 39238 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 32 T^{2} + 760 T^{3} + 18046 T^{4} + 760 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.54194311199417550720019623236, −5.51400583549458295656811837794, −5.30573431360074854962504127005, −5.10093991843398191310918141067, −5.09614184092633099521472023471, −4.98216671608052652283445951302, −4.79123454200317900956382310572, −4.71526499195494545692205241163, −4.13305227326805171179908027124, −4.12672430378085787190694585318, −4.07078090330073732339788102932, −3.89169995707503292765480362264, −3.51414119928654478589056970180, −3.46671434856337795698064935359, −3.29703875223300126537358009325, −3.07775458957219750842948249413, −2.87619941409330903017530824156, −2.87171723637115852019108995166, −2.83191318942971110173199148259, −2.34428497374263965322526383139, −2.17309780129609237942546923878, −1.92304072184182163364852366611, −1.66293601948795390577150572353, −1.38672661548188043473243121869, −0.68364794438649643120869448681,
0.68364794438649643120869448681, 1.38672661548188043473243121869, 1.66293601948795390577150572353, 1.92304072184182163364852366611, 2.17309780129609237942546923878, 2.34428497374263965322526383139, 2.83191318942971110173199148259, 2.87171723637115852019108995166, 2.87619941409330903017530824156, 3.07775458957219750842948249413, 3.29703875223300126537358009325, 3.46671434856337795698064935359, 3.51414119928654478589056970180, 3.89169995707503292765480362264, 4.07078090330073732339788102932, 4.12672430378085787190694585318, 4.13305227326805171179908027124, 4.71526499195494545692205241163, 4.79123454200317900956382310572, 4.98216671608052652283445951302, 5.09614184092633099521472023471, 5.10093991843398191310918141067, 5.30573431360074854962504127005, 5.51400583549458295656811837794, 5.54194311199417550720019623236
Plot not available for L-functions of degree greater than 10.
