| L(s) = 1 | − 2·16-s + 56·31-s + 40·61-s + 88·121-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 + ⋯ |
| L(s) = 1 | − 1/2·16-s + 10.0·31-s + 5.12·61-s + 8·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.18266703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.18266703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 958 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 7 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 2737 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + 3191 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 4174 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 5 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 8302 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10657 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 3122 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 4657 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−3.98166576338394932210969768525, −3.97105957240062381972839336065, −3.92495058109933924548194029437, −3.85623985475172646139031386559, −3.69710165835026328341482135681, −3.19627476439032348936910898264, −3.15625278893659544408261202372, −3.15164163535647437438644188656, −3.04121627569113482079975580311, −2.97404724489201598206639185458, −2.80991711358673535395123777866, −2.55043497088047339196635734269, −2.39636281011616177037460258591, −2.38724638186623045499979595607, −2.38526298648098267894473480954, −2.05420660940367621869200611317, −1.94094925174734476547812087214, −1.74389751735763430914934563273, −1.38822748557989546825409967720, −1.31053673216358729449850708030, −0.963472459348031127707773569195, −0.904189285378749443931368489149, −0.77669000650196337738542581241, −0.53038890879752740608558706836, −0.52086423632855374155832590026,
0.52086423632855374155832590026, 0.53038890879752740608558706836, 0.77669000650196337738542581241, 0.904189285378749443931368489149, 0.963472459348031127707773569195, 1.31053673216358729449850708030, 1.38822748557989546825409967720, 1.74389751735763430914934563273, 1.94094925174734476547812087214, 2.05420660940367621869200611317, 2.38526298648098267894473480954, 2.38724638186623045499979595607, 2.39636281011616177037460258591, 2.55043497088047339196635734269, 2.80991711358673535395123777866, 2.97404724489201598206639185458, 3.04121627569113482079975580311, 3.15164163535647437438644188656, 3.15625278893659544408261202372, 3.19627476439032348936910898264, 3.69710165835026328341482135681, 3.85623985475172646139031386559, 3.92495058109933924548194029437, 3.97105957240062381972839336065, 3.98166576338394932210969768525
Plot not available for L-functions of degree greater than 10.
