| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s + 2·9-s − 2·11-s + 330·16-s − 16·18-s + 16·22-s − 6·23-s + 9·25-s + 16·29-s − 792·32-s + 72·36-s − 24·37-s − 10·43-s − 72·44-s + 48·46-s − 72·50-s + 16·53-s − 128·58-s + 1.71e3·64-s − 4·67-s − 44·71-s − 240·72-s + 192·74-s + 68·79-s − 15·81-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s + 2/3·9-s − 0.603·11-s + 82.5·16-s − 3.77·18-s + 3.41·22-s − 1.25·23-s + 9/5·25-s + 2.97·29-s − 140.·32-s + 12·36-s − 3.94·37-s − 1.52·43-s − 10.8·44-s + 7.07·46-s − 10.1·50-s + 2.19·53-s − 16.8·58-s + 214.5·64-s − 0.488·67-s − 5.22·71-s − 28.2·72-s + 22.3·74-s + 7.65·79-s − 5/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2303548567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2303548567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 9 T^{2} + 37 T^{4} + 54 T^{6} - 714 T^{8} + 54 p^{2} T^{10} + 37 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + T + 5 T^{2} - 26 T^{3} - 116 T^{4} - 26 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 8 T^{2} + 10 p T^{4} + 3232 T^{6} - 35021 T^{8} + 3232 p^{2} T^{10} + 10 p^{5} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2}( 1 + 29 T^{2} + 552 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 19 | \( ( 1 - 31 T^{2} + 600 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 3 T - 13 T^{2} - 72 T^{3} - 252 T^{4} - 72 p T^{5} - 13 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 80 T^{2} + 3102 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 125 T^{2} + 8593 T^{4} - 458750 T^{6} + 20357638 T^{8} - 458750 p^{2} T^{10} + 8593 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 5 T - 41 T^{2} - 100 T^{3} + 1432 T^{4} - 100 p T^{5} - 41 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 135 T^{2} + 10232 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 173 T^{2} + 14688 T^{4} + 173 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + T + 108 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 11 T + 146 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 113 T^{2} + 205 T^{4} - 215378 T^{6} + 61409854 T^{8} - 215378 p^{2} T^{10} + 205 p^{4} T^{12} - 113 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 17 T + 204 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 106 T^{2} + 4347 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 72 T^{2} - 8174 T^{4} + 178848 T^{6} + 74631459 T^{8} + 178848 p^{2} T^{10} - 8174 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 113 T^{2} + 7165 T^{4} + 1493182 T^{6} - 177267986 T^{8} + 1493182 p^{2} T^{10} + 7165 p^{4} T^{12} - 113 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.36997651630912703375951963203, −4.29685521584255833564895878593, −3.79795032214709100630806276309, −3.75957141432602261155940763625, −3.75728888570017462451493557143, −3.57737959912353402083533233127, −3.31991201948301695059288373010, −3.30108125470873898094082072887, −3.01316554122317673620353642721, −2.92955349750850534416840782887, −2.82005234865891262477698771919, −2.81539232907033730706370063179, −2.47009987677257503743437000964, −2.19085654272500259806547362471, −2.16736684199961470448634858753, −2.09715210689817904472746417727, −1.84237894558829012693916298048, −1.76904589034070601368683460997, −1.46964047769840073600813655097, −1.37450000952811111801268938051, −1.11337212419518333085135136463, −1.01972706991935924830503262835, −0.62457891773606744016572001275, −0.45810423527242280440396446177, −0.30423639418784810620121974049,
0.30423639418784810620121974049, 0.45810423527242280440396446177, 0.62457891773606744016572001275, 1.01972706991935924830503262835, 1.11337212419518333085135136463, 1.37450000952811111801268938051, 1.46964047769840073600813655097, 1.76904589034070601368683460997, 1.84237894558829012693916298048, 2.09715210689817904472746417727, 2.16736684199961470448634858753, 2.19085654272500259806547362471, 2.47009987677257503743437000964, 2.81539232907033730706370063179, 2.82005234865891262477698771919, 2.92955349750850534416840782887, 3.01316554122317673620353642721, 3.30108125470873898094082072887, 3.31991201948301695059288373010, 3.57737959912353402083533233127, 3.75728888570017462451493557143, 3.75957141432602261155940763625, 3.79795032214709100630806276309, 4.29685521584255833564895878593, 4.36997651630912703375951963203
Plot not available for L-functions of degree greater than 10.
