| L(s) = 1 | + 56·49-s − 40·81-s − 176·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | + 8·49-s − 4.44·81-s − 16·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07722330166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07722330166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 22 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - p T^{2} )^{8} \) |
| good | 3 | \( ( 1 + 10 T^{4} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 + p T^{2} )^{16} \) |
| 13 | \( ( 1 - 310 T^{4} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 + p T^{2} )^{16} \) |
| 19 | \( ( 1 + 650 T^{4} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{8} \) |
| 29 | \( ( 1 - p T^{2} )^{16} \) |
| 31 | \( ( 1 + p T^{2} )^{16} \) |
| 37 | \( ( 1 + p T^{2} )^{16} \) |
| 41 | \( ( 1 - p T^{2} )^{16} \) |
| 43 | \( ( 1 - p T^{2} )^{16} \) |
| 47 | \( ( 1 - p T^{2} )^{16} \) |
| 53 | \( ( 1 + p T^{2} )^{16} \) |
| 59 | \( ( 1 + 1130 T^{4} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 + 7370 T^{4} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - p T^{2} )^{16} \) |
| 71 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{8} \) |
| 73 | \( ( 1 + p T^{2} )^{16} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 + 13130 T^{4} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - p T^{2} )^{16} \) |
| 97 | \( ( 1 + p T^{2} )^{16} \) |
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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
−2.24326225286291344652739758984, −2.20291327126239517998013240128, −2.13162324366914990745193651781, −2.08933638131478247430689577902, −2.06242476762599219700006384843, −1.97153314519423423621693039257, −1.68312426272319101630890056512, −1.60872514533390451626910844631, −1.54453802420477711684025711678, −1.50380259248298829023981200663, −1.49081555825325346608014738486, −1.44344522957845412671821676380, −1.42702263101298294303620256823, −1.22634152846036850955283407651, −1.13667687268288245940607972511, −1.04310509717718310794975407130, −0.969931827526012026402293686003, −0.924653059221757662986693068396, −0.855290623212016635208127456367, −0.74248320143640227645291254617, −0.45026801616159895613976808015, −0.44425301532945436533360114095, −0.38419237923052095631232399142, −0.19230658272492545818410625937, −0.01343023243779722351031592582,
0.01343023243779722351031592582, 0.19230658272492545818410625937, 0.38419237923052095631232399142, 0.44425301532945436533360114095, 0.45026801616159895613976808015, 0.74248320143640227645291254617, 0.855290623212016635208127456367, 0.924653059221757662986693068396, 0.969931827526012026402293686003, 1.04310509717718310794975407130, 1.13667687268288245940607972511, 1.22634152846036850955283407651, 1.42702263101298294303620256823, 1.44344522957845412671821676380, 1.49081555825325346608014738486, 1.50380259248298829023981200663, 1.54453802420477711684025711678, 1.60872514533390451626910844631, 1.68312426272319101630890056512, 1.97153314519423423621693039257, 2.06242476762599219700006384843, 2.08933638131478247430689577902, 2.13162324366914990745193651781, 2.20291327126239517998013240128, 2.24326225286291344652739758984
Plot not available for L-functions of degree greater than 10.
