| L(s) = 1 | − 4·5-s + 46·9-s − 44·11-s − 8·25-s − 52·31-s − 184·45-s − 54·49-s + 176·55-s + 384·59-s − 60·71-s + 1.01e3·81-s + 380·89-s − 2.02e3·99-s + 808·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 208·155-s + 157-s + 163-s + 167-s − 80·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4/5·5-s + 46/9·9-s − 4·11-s − 0.319·25-s − 1.67·31-s − 4.08·45-s − 1.10·49-s + 16/5·55-s + 6.50·59-s − 0.845·71-s + 12.5·81-s + 4.26·89-s − 20.4·99-s + 6.67·121-s − 0.0959·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.34·155-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.473·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.316147971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.316147971\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 2 T + 2 p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | \( ( 1 + 2 p T + 322 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 23 T^{2} + 284 T^{4} - 23 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 27 T^{2} + 464 T^{4} + 27 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 40 T^{2} - 1682 T^{4} + 40 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 987 T^{2} + 406064 T^{4} + 987 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 21 T^{2} + 248196 T^{4} + 21 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1520 T^{2} + 1050526 T^{4} - 1520 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 483 T^{2} - 78156 T^{4} - 483 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 13 T + 1708 T^{2} + 13 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 3279 T^{2} + 5230380 T^{4} - 3279 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 864 T^{2} + 5637246 T^{4} - 864 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 900 T^{2} + 6561878 T^{4} + 900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 7536 T^{2} + 23854686 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4607 T^{2} + 20681164 T^{4} - 4607 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 96 T + 9102 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 9683 T^{2} + 44390164 T^{4} - 9683 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 13968 T^{2} + 85339134 T^{4} - 13968 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 15 T - 1024 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4900 T^{2} + 971638 T^{4} + 4900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2880 T^{2} + 43900158 T^{4} - 2880 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 19320 T^{2} + 175466318 T^{4} + 19320 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 95 T + 12676 T^{2} - 95 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16784 T^{2} + 247121950 T^{4} - 16784 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−5.12371190304387059463766355879, −5.06503978111373910170157559101, −4.96065953205525262277703838506, −4.95817495376257355266677803928, −4.76912383762014431023081317072, −4.46516632303229604491152943256, −4.33041499141730556947480921504, −4.31113142585153098827172972034, −3.85919827550093231703332306514, −3.80959503583110417114631049030, −3.68923676461439834450214783087, −3.65964288061101095932823763069, −3.60093117925484231279984358803, −3.05635625350736155302660111018, −2.87925112735997183404711953037, −2.69852615791016070148328551922, −2.24720559088383407680425799625, −2.22567999093016929061131440368, −2.08832238849252164983301472856, −2.03738123438367153917697316449, −1.44540961268259782242791662437, −1.23174550523228489123105114480, −1.12108460508023739964483058316, −0.53336027156550841545320818839, −0.31080911209359190479379110745,
0.31080911209359190479379110745, 0.53336027156550841545320818839, 1.12108460508023739964483058316, 1.23174550523228489123105114480, 1.44540961268259782242791662437, 2.03738123438367153917697316449, 2.08832238849252164983301472856, 2.22567999093016929061131440368, 2.24720559088383407680425799625, 2.69852615791016070148328551922, 2.87925112735997183404711953037, 3.05635625350736155302660111018, 3.60093117925484231279984358803, 3.65964288061101095932823763069, 3.68923676461439834450214783087, 3.80959503583110417114631049030, 3.85919827550093231703332306514, 4.31113142585153098827172972034, 4.33041499141730556947480921504, 4.46516632303229604491152943256, 4.76912383762014431023081317072, 4.95817495376257355266677803928, 4.96065953205525262277703838506, 5.06503978111373910170157559101, 5.12371190304387059463766355879
Plot not available for L-functions of degree greater than 10.
