| L(s) = 1 | − 4·2-s + 6·4-s − 4·11-s − 15·16-s + 16·22-s − 12·23-s − 18·25-s − 16·29-s + 24·32-s − 24·37-s − 10·43-s − 24·44-s + 48·46-s + 72·50-s − 16·53-s + 64·58-s − 6·64-s + 2·67-s + 44·71-s + 96·74-s − 34·79-s + 40·86-s − 72·92-s − 108·100-s + 64·106-s + 22·107-s − 4·109-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 1.20·11-s − 3.75·16-s + 3.41·22-s − 2.50·23-s − 3.59·25-s − 2.97·29-s + 4.24·32-s − 3.94·37-s − 1.52·43-s − 3.61·44-s + 7.07·46-s + 10.1·50-s − 2.19·53-s + 8.40·58-s − 3/4·64-s + 0.244·67-s + 5.22·71-s + 11.1·74-s − 3.82·79-s + 4.31·86-s − 7.50·92-s − 10.7·100-s + 6.21·106-s + 2.12·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04550219393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04550219393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 9 T^{2} + 44 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 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 | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 80 T^{2} + 3298 T^{4} - 94400 T^{6} + 2548483 T^{8} - 94400 p^{2} T^{10} + 3298 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( ( 1 + 18 T^{2} - 1885 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 135 T^{2} + 7993 T^{4} + 441450 T^{6} + 29135238 T^{8} + 441450 p^{2} T^{10} + 7993 p^{4} T^{12} + 135 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 173 T^{2} + 15241 T^{4} - 1253558 T^{6} + 90525694 T^{8} - 1253558 p^{2} T^{10} + 15241 p^{4} T^{12} - 173 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - T - 107 T^{2} + 26 T^{3} + 7108 T^{4} + 26 p T^{5} - 107 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( ( 1 + 17 T + 85 T^{2} + 782 T^{3} + 12544 T^{4} + 782 p T^{5} + 85 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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
−3.65486542283708923995104698270, −3.54253562614084696079683756083, −3.43027974537344392471221035292, −3.29451252065673083100598323492, −3.24382434724828319399101704310, −3.14307586919654715347231793409, −3.07124423888081159938115425376, −2.73449695963466005612677415678, −2.41339308197948482830687826391, −2.38996781275699422646323398663, −2.36274910439347083200221068899, −2.31974166491121168735948560706, −1.95029454977668915389949119229, −1.87527306326502009730232128928, −1.85759244927849902892385370747, −1.76157337641372659656382962143, −1.59803066949590523785401569170, −1.54504677672543088888453076851, −1.31845089179623507615763830680, −1.28896361913515283087969458896, −0.69154218078514815447702491241, −0.53494906639995368768937372957, −0.39871151090303408056122523740, −0.32287934620644146823539808845, −0.094160489831582499729300601183,
0.094160489831582499729300601183, 0.32287934620644146823539808845, 0.39871151090303408056122523740, 0.53494906639995368768937372957, 0.69154218078514815447702491241, 1.28896361913515283087969458896, 1.31845089179623507615763830680, 1.54504677672543088888453076851, 1.59803066949590523785401569170, 1.76157337641372659656382962143, 1.85759244927849902892385370747, 1.87527306326502009730232128928, 1.95029454977668915389949119229, 2.31974166491121168735948560706, 2.36274910439347083200221068899, 2.38996781275699422646323398663, 2.41339308197948482830687826391, 2.73449695963466005612677415678, 3.07124423888081159938115425376, 3.14307586919654715347231793409, 3.24382434724828319399101704310, 3.29451252065673083100598323492, 3.43027974537344392471221035292, 3.54253562614084696079683756083, 3.65486542283708923995104698270
Plot not available for L-functions of degree greater than 10.
