| L(s) = 1 | − 26·9-s − 408·19-s − 1.52e3·31-s − 9.84e3·49-s − 1.08e4·61-s − 1.51e4·79-s + 9.01e3·81-s − 2.81e4·109-s + 5.92e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.42e5·169-s + 1.06e4·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 0.320·9-s − 1.13·19-s − 1.59·31-s − 4.09·49-s − 2.90·61-s − 2.42·79-s + 1.37·81-s − 2.37·109-s + 0.404·121-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 − 4.99·169-s + 0.362·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01541905992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01541905992\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 26 T^{2} - 926 p^{2} T^{4} + 26 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 4922 T^{2} + 5298 p^{4} T^{4} + 4922 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2964 T^{2} + 418458086 T^{4} - 2964 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 13 | \( ( 1 + 71332 T^{2} + 2455108998 T^{4} + 71332 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 233184 T^{2} + 26236550846 T^{4} - 233184 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 + 102 T + 68618 T^{2} + 102 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( ( 1 - 823014 T^{2} + 325413279986 T^{4} - 823014 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 - 762724 T^{2} + 1109608104966 T^{4} - 762724 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 31 | \( ( 1 + 382 T + 1688898 T^{2} + 382 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 37 | \( ( 1 - 702844 T^{2} + 6115305865926 T^{4} - 702844 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 - 3942144 T^{2} + 17357795628926 T^{4} - 3942144 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 43 | \( ( 1 + 12299642 T^{2} + 61110112365618 T^{4} + 12299642 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4189626 T^{2} + 48598849986866 T^{4} + 4189626 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16577824 T^{2} + 158163732867966 T^{4} - 16577824 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 26333844 T^{2} + 385333281386726 T^{4} - 26333844 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2702 T + 19980258 T^{2} + 2702 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 67 | \( ( 1 - 26275078 T^{2} + 846492858320178 T^{4} - 26275078 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 76389124 T^{2} + 2705554600767366 T^{4} - 76389124 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 + 75060836 T^{2} + 2876952445826886 T^{4} + 75060836 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 3786 T + 79731986 T^{2} + 3786 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 83 | \( ( 1 - 159928134 T^{2} + 10738782650176946 T^{4} - 159928134 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 145506564 T^{2} + 13157090693559686 T^{4} - 145506564 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 185051492 T^{2} + 23317632065034438 T^{4} + 185051492 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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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
−4.54839460649564191612512591862, −4.52249813913687608622299412875, −4.38646054478520924510705855967, −3.80510783236842460015825317415, −3.76912598887598897949006945576, −3.67885593113737474916016416089, −3.65376517777248193824929715992, −3.57026030074490303921930644391, −3.40042665384527287711321128227, −3.13875304107387171541050841552, −2.84719774138069796060600879335, −2.64707566970545245943150266616, −2.45364045370485471145044287915, −2.40616456181386283599612137196, −2.33868630226238710719174240120, −2.21301128131782347584321982888, −1.58987770407451360016763044287, −1.58041552842042862051348860756, −1.33720403365478192133150132363, −1.32135075240677117171793219476, −1.20032804154912541815768774930, −0.967669337343914492410518369586, −0.23782330639513417838028731660, −0.11698178914349341305108491486, −0.05167422049679418271818991189,
0.05167422049679418271818991189, 0.11698178914349341305108491486, 0.23782330639513417838028731660, 0.967669337343914492410518369586, 1.20032804154912541815768774930, 1.32135075240677117171793219476, 1.33720403365478192133150132363, 1.58041552842042862051348860756, 1.58987770407451360016763044287, 2.21301128131782347584321982888, 2.33868630226238710719174240120, 2.40616456181386283599612137196, 2.45364045370485471145044287915, 2.64707566970545245943150266616, 2.84719774138069796060600879335, 3.13875304107387171541050841552, 3.40042665384527287711321128227, 3.57026030074490303921930644391, 3.65376517777248193824929715992, 3.67885593113737474916016416089, 3.76912598887598897949006945576, 3.80510783236842460015825317415, 4.38646054478520924510705855967, 4.52249813913687608622299412875, 4.54839460649564191612512591862
Plot not available for L-functions of degree greater than 10.
