| L(s) = 1 | + 8·2-s + 36·4-s + 120·8-s − 5·9-s + 330·16-s − 40·18-s + 6·25-s + 792·32-s − 180·36-s − 44·43-s + 48·50-s + 1.71e3·64-s − 16·71-s − 600·72-s − 52·79-s + 9·81-s − 352·86-s + 216·100-s + 44·121-s + 127-s + 3.43e3·128-s + 131-s + 137-s + 139-s − 128·142-s − 1.65e3·144-s + 149-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s + 42.4·8-s − 5/3·9-s + 82.5·16-s − 9.42·18-s + 6/5·25-s + 140.·32-s − 30·36-s − 6.70·43-s + 6.78·50-s + 214.5·64-s − 1.89·71-s − 70.7·72-s − 5.85·79-s + 81-s − 37.9·86-s + 21.5·100-s + 4·121-s + 0.0887·127-s + 303.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·142-s − 137.5·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(85.69306267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(85.69306267\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 3 T^{2} + 44 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 61 T^{2} + 1500 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 13 T^{2} + 96 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 p T^{2} + 1059 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 53 T^{2} + 1716 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 73 T^{2} + 3180 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 54 T^{2} + 2939 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 87 T^{2} + 4256 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 59 T^{2} + 3432 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 152 T^{2} + 11550 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 170 T^{2} + 14139 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 225 T^{2} + 21428 T^{4} - 225 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 174 T^{2} + 16115 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 13 T + 192 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 264 T^{2} + 31070 T^{4} - 264 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 + 249 T^{2} + 34112 T^{4} + 249 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.73322594685884262515154529274, −4.68760980684225302994677591388, −4.37973124516603654427053034982, −4.32055677697546549048207607212, −4.26526917236577712319365534431, −3.98390049350246380385482696181, −3.93050173832710218836571642714, −3.83109494055270580110939095905, −3.72681711696237823516420305812, −3.41686910836999343846367094809, −3.36138237782426975045777791441, −3.08042349620092918697449558194, −3.05513769817283490848281531666, −2.90725762617720956931240085978, −2.83984808060290603046460499705, −2.81276576495960604544452290902, −2.60157773487474276754149946894, −2.41077527858032448449584408337, −2.03513619317648076262665409234, −1.84292895993510538927001192635, −1.71162671381079151204383656415, −1.57188489034620111814591145362, −1.38180897512599059992427327185, −1.19691416409276258089308247251, −0.27864259377516250728668501508,
0.27864259377516250728668501508, 1.19691416409276258089308247251, 1.38180897512599059992427327185, 1.57188489034620111814591145362, 1.71162671381079151204383656415, 1.84292895993510538927001192635, 2.03513619317648076262665409234, 2.41077527858032448449584408337, 2.60157773487474276754149946894, 2.81276576495960604544452290902, 2.83984808060290603046460499705, 2.90725762617720956931240085978, 3.05513769817283490848281531666, 3.08042349620092918697449558194, 3.36138237782426975045777791441, 3.41686910836999343846367094809, 3.72681711696237823516420305812, 3.83109494055270580110939095905, 3.93050173832710218836571642714, 3.98390049350246380385482696181, 4.26526917236577712319365534431, 4.32055677697546549048207607212, 4.37973124516603654427053034982, 4.68760980684225302994677591388, 4.73322594685884262515154529274
Plot not available for L-functions of degree greater than 10.
