| L(s) = 1 | + 8·13-s − 8·37-s − 8·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 | + 8·13-s − 8·37-s − 8·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.937859458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937859458\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T^{8} \) |
| good | 3 | \( 1 + T^{16} \) |
| 5 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{2} )^{8} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.68409307183499473672401296016, −3.61794003478538298139978442845, −3.60153553870245231910924115158, −3.48887547597368530991921859565, −3.46540383603642607143530869685, −3.33368511260173386938354206174, −3.17918083295035340943181955500, −3.05781310144899554919331116070, −2.98650488985074002478277665291, −2.81865317324375542194820081394, −2.68242428109627475267723014912, −2.52400796067588744735983381558, −2.32936877420768256629508959265, −1.97423523826817961495961025694, −1.86896000179251455647016839950, −1.76327183261038844085490232735, −1.74653701624830739458881487962, −1.74538984739385140886013182202, −1.45979617298016292532371462923, −1.43012057307323231097592152331, −1.29933171558254912411907023465, −1.26756853036687390816349607319, −0.965159664322735468485494489528, −0.71497100138317770462887077094, −0.31048825321344191085752977878,
0.31048825321344191085752977878, 0.71497100138317770462887077094, 0.965159664322735468485494489528, 1.26756853036687390816349607319, 1.29933171558254912411907023465, 1.43012057307323231097592152331, 1.45979617298016292532371462923, 1.74538984739385140886013182202, 1.74653701624830739458881487962, 1.76327183261038844085490232735, 1.86896000179251455647016839950, 1.97423523826817961495961025694, 2.32936877420768256629508959265, 2.52400796067588744735983381558, 2.68242428109627475267723014912, 2.81865317324375542194820081394, 2.98650488985074002478277665291, 3.05781310144899554919331116070, 3.17918083295035340943181955500, 3.33368511260173386938354206174, 3.46540383603642607143530869685, 3.48887547597368530991921859565, 3.60153553870245231910924115158, 3.61794003478538298139978442845, 3.68409307183499473672401296016
Plot not available for L-functions of degree greater than 10.
