| L(s) = 1 | + 5·4-s − 12·9-s + 16-s + 184·29-s − 60·36-s − 256·41-s + 208·49-s + 304·61-s − 35·64-s + 90·81-s − 560·89-s + 296·101-s + 608·109-s + 920·116-s + 272·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s − 1.28e3·164-s + 167-s − 1.18e3·169-s + ⋯ |
| L(s) = 1 | + 5/4·4-s − 4/3·9-s + 1/16·16-s + 6.34·29-s − 5/3·36-s − 6.24·41-s + 4.24·49-s + 4.98·61-s − 0.546·64-s + 10/9·81-s − 6.29·89-s + 2.93·101-s + 5.57·109-s + 7.93·116-s + 2.24·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.0833·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 7.80·164-s + 0.00598·167-s − 7.00·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.127897911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.127897911\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5 T^{2} + 3 p^{3} T^{4} - 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 104 T^{2} + 5454 T^{4} - 104 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 136 T^{2} + 28206 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 592 T^{2} + 144510 T^{4} + 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 112 T^{2} + 118878 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 436 T^{2} + 275334 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 76 p T^{2} + 1290726 T^{4} - 76 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 46 T + 1698 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1540 T^{2} + 1506054 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4720 T^{2} + 9299454 T^{4} + 4720 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 64 T + 2334 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 4868 T^{2} + 12528486 T^{4} - 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 5540 T^{2} + 15331014 T^{4} - 5540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 10480 T^{2} + 43220094 T^{4} + 10480 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 2744 T^{2} + 25695534 T^{4} + 2744 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 38 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 - 8564 T^{2} + 44158854 T^{4} - 8564 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3028 T^{2} - 19410330 T^{4} - 3028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 17140 T^{2} + 129420582 T^{4} + 17140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 22660 T^{2} + 205335174 T^{4} - 22660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 23492 T^{2} + 230783910 T^{4} - 23492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 140 T + 19830 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 8252 T^{2} + 163205766 T^{4} - 8252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.89376171146050178582818146254, −4.80265910588590143078352346866, −4.72980561476101088613733249221, −4.67032171878307283746176270561, −4.66960421430614309896562343949, −4.16063810637871996191045803163, −3.97503250161900425941027274569, −3.89879478237775251643107594823, −3.78085566632352833159707844003, −3.50878674978183564368675763170, −3.34198862763657451562741671971, −3.11295419370734816401541656864, −3.03805990735797327262258074952, −2.93752835409112067014494226337, −2.56148364933860140688152000392, −2.53048211116850676271119731807, −2.34414731753038604430487325307, −2.23182389557541751073299093947, −2.00451979008810136273486252635, −1.71287557504209260346883154989, −1.30712248989769197196717971695, −1.19896048520073346350414615201, −0.74513416850080723783452993718, −0.70807521980678229925563562651, −0.29852556652790469363851623908,
0.29852556652790469363851623908, 0.70807521980678229925563562651, 0.74513416850080723783452993718, 1.19896048520073346350414615201, 1.30712248989769197196717971695, 1.71287557504209260346883154989, 2.00451979008810136273486252635, 2.23182389557541751073299093947, 2.34414731753038604430487325307, 2.53048211116850676271119731807, 2.56148364933860140688152000392, 2.93752835409112067014494226337, 3.03805990735797327262258074952, 3.11295419370734816401541656864, 3.34198862763657451562741671971, 3.50878674978183564368675763170, 3.78085566632352833159707844003, 3.89879478237775251643107594823, 3.97503250161900425941027274569, 4.16063810637871996191045803163, 4.66960421430614309896562343949, 4.67032171878307283746176270561, 4.72980561476101088613733249221, 4.80265910588590143078352346866, 4.89376171146050178582818146254
Plot not available for L-functions of degree greater than 10.
