| L(s) = 1 | + 504·9-s − 560·25-s + 7.05e3·41-s − 2.31e4·49-s + 1.21e5·81-s − 4.03e4·89-s − 1.05e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.43e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.82e5·225-s + ⋯ |
| L(s) = 1 | + 56/9·9-s − 0.895·25-s + 4.19·41-s − 9.63·49-s + 18.5·81-s − 5.09·89-s − 7.17·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 15.5·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s − 5.57·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.02630003709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02630003709\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 56 p T^{2} + 894 p^{2} T^{4} + 56 p^{9} T^{6} + p^{16} T^{8} )^{2} \) |
| good | 3 | \( ( 1 - 14 p^{2} T^{2} + 1034 p^{2} T^{4} - 14 p^{10} T^{6} + p^{16} T^{8} )^{4} \) |
| 7 | \( ( 1 + 118 p^{2} T^{2} + 398082 p^{2} T^{4} + 118 p^{10} T^{6} + p^{16} T^{8} )^{4} \) |
| 11 | \( ( 1 + 26252 T^{2} + 503354598 T^{4} + 26252 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 13 | \( ( 1 + 110992 T^{2} + 4711236318 T^{4} + 110992 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 17 | \( ( 1 + 25604 T^{2} + 11912064246 T^{4} + 25604 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 19 | \( ( 1 + 125356 T^{2} + 32084186406 T^{4} + 125356 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 23 | \( ( 1 + 645142 T^{2} + 204717301218 T^{4} + 645142 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 29 | \( ( 1 - 1144532 T^{2} + 679332011238 T^{4} - 1144532 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 31 | \( ( 1 - 500852 T^{2} + 1336640255718 T^{4} - 500852 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 37 | \( ( 1 + 5060056 T^{2} + 12010965326286 T^{4} + 5060056 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 41 | \( ( 1 - 882 T + 2412818 T^{2} - 882 p^{4} T^{3} + p^{8} T^{4} )^{8} \) |
| 43 | \( ( 1 - 148474 p T^{2} + 28815779478618 T^{4} - 148474 p^{9} T^{6} + p^{16} T^{8} )^{4} \) |
| 47 | \( ( 1 + 2624182 T^{2} + 25986815750178 T^{4} + 2624182 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 53 | \( ( 1 - 3340496 T^{2} + 64516575165726 T^{4} - 3340496 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 59 | \( ( 1 + 23229164 T^{2} + 412028982819366 T^{4} + 23229164 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 61 | \( ( 1 - 16851016 T^{2} + 454223995956366 T^{4} - 16851016 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 67 | \( ( 1 - 13608142 T^{2} + 825459111750138 T^{4} - 13608142 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 71 | \( ( 1 - 87601972 T^{2} + 3209588496287718 T^{4} - 87601972 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 73 | \( ( 1 - 70053244 T^{2} + 2370886281245046 T^{4} - 70053244 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 79 | \( ( 1 - 136259444 T^{2} + 7585408400210406 T^{4} - 136259444 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 83 | \( ( 1 + 63289058 T^{2} + 3995830538960538 T^{4} + 63289058 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
| 89 | \( ( 1 + 5040 T + 118102142 T^{2} + 5040 p^{4} T^{3} + p^{8} T^{4} )^{8} \) |
| 97 | \( ( 1 - 70173436 T^{2} + 271861370899446 T^{4} - 70173436 p^{8} T^{6} + p^{16} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.36990243306759566839531834649, −2.35979104749175900039824227522, −2.30362626005925303403228528777, −2.27677782150249935504754537706, −2.12608172627289061464276425601, −1.95191538348398790248723558236, −1.83824190936495353633710253857, −1.77643463656736453822911726945, −1.62280848798819634516335215107, −1.61694106586521940148820023147, −1.49005868570435388967715887515, −1.36906857610083685461617994293, −1.35090818568576955830557731098, −1.24603616141482510135990232664, −1.19863246979872450091219706104, −1.16218745115838365041974932617, −1.12676210626886646698704092529, −1.08576265746140746657067955910, −0.875343809891754341091535750489, −0.59682127879864068823858687506, −0.41528389631339930826618661785, −0.30351218255978137764356783256, −0.27327512082860094963710414920, −0.25785651427079523977287780810, −0.00428506455268422826333378982,
0.00428506455268422826333378982, 0.25785651427079523977287780810, 0.27327512082860094963710414920, 0.30351218255978137764356783256, 0.41528389631339930826618661785, 0.59682127879864068823858687506, 0.875343809891754341091535750489, 1.08576265746140746657067955910, 1.12676210626886646698704092529, 1.16218745115838365041974932617, 1.19863246979872450091219706104, 1.24603616141482510135990232664, 1.35090818568576955830557731098, 1.36906857610083685461617994293, 1.49005868570435388967715887515, 1.61694106586521940148820023147, 1.62280848798819634516335215107, 1.77643463656736453822911726945, 1.83824190936495353633710253857, 1.95191538348398790248723558236, 2.12608172627289061464276425601, 2.27677782150249935504754537706, 2.30362626005925303403228528777, 2.35979104749175900039824227522, 2.36990243306759566839531834649
Plot not available for L-functions of degree greater than 10.
