| L(s) = 1 | − 8·11-s − 2·16-s + 8·19-s − 24·49-s + 40·59-s + 40·71-s − 2·81-s + 24·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 16·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 64·209-s + 211-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 1/2·16-s + 1.83·19-s − 3.42·49-s + 5.20·59-s + 4.74·71-s − 2/9·81-s + 2.54·89-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 1.20·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 4.42·209-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7087884714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7087884714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 292 T^{4} + 48486 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 - 252 T^{4} + 99718 T^{8} - 252 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 252 T^{4} + 242758 T^{8} - 252 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 - 12 T^{2} + 1126 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1294 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 40 T^{2} + 1214 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 188 T^{4} - 5134362 T^{8} - 188 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 4068 T^{4} + 12565318 T^{8} + 4068 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 1692 T^{4} - 2725850 T^{8} - 1692 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 24 T^{2} + 1294 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 2492 T^{4} - 10980570 T^{8} - 2492 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 388 T^{4} - 43924410 T^{8} + 388 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 140 T^{2} + 9894 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 13294 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2}( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−4.21091779778514945593013136242, −4.19635030757769315092824705999, −3.90650818890741261987671273654, −3.77855427399228528488934186387, −3.73095692798268406559716613412, −3.58084970471856343976444472332, −3.56396219960671085786419231527, −3.33553295863975297319403707125, −3.03307339467208135748382819492, −3.03031001505931980946117789777, −2.91212056696836533807644943293, −2.73957903803458211559775492663, −2.61756621266987405887390248690, −2.42960699275739383497515183985, −2.28953214516944944050779200134, −2.13406849778631968040534368330, −2.10343239680323526559864042410, −1.81030225828598025319806012801, −1.76057992767487492933421105300, −1.28351249459018792277211358631, −1.22814018437096173803351630520, −0.925003527566161802898413571642, −0.834610592128161244953037547353, −0.50246723366873247310050625418, −0.11375928582631120710693729334,
0.11375928582631120710693729334, 0.50246723366873247310050625418, 0.834610592128161244953037547353, 0.925003527566161802898413571642, 1.22814018437096173803351630520, 1.28351249459018792277211358631, 1.76057992767487492933421105300, 1.81030225828598025319806012801, 2.10343239680323526559864042410, 2.13406849778631968040534368330, 2.28953214516944944050779200134, 2.42960699275739383497515183985, 2.61756621266987405887390248690, 2.73957903803458211559775492663, 2.91212056696836533807644943293, 3.03031001505931980946117789777, 3.03307339467208135748382819492, 3.33553295863975297319403707125, 3.56396219960671085786419231527, 3.58084970471856343976444472332, 3.73095692798268406559716613412, 3.77855427399228528488934186387, 3.90650818890741261987671273654, 4.19635030757769315092824705999, 4.21091779778514945593013136242
Plot not available for L-functions of degree greater than 10.
