| L(s) = 1 | − 4.84e4·31-s + 2.33e5·61-s + 3.16e4·81-s + 6.89e5·121-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 | − 9.05·31-s + 8.04·61-s + 0.536·81-s + 4.28·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + 1.20e−6·233-s + 1.13e−6·239-s + 1.10e−6·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.821009508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.821009508\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3518 p^{2} T^{4} + p^{20} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 457164574 T^{4} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 - 172342 T^{2} + p^{10} T^{4} )^{4} \) |
| 13 | \( ( 1 + 257618185106 T^{4} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 + 3873297949058 T^{4} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4127734 T^{2} + p^{10} T^{4} )^{4} \) |
| 23 | \( ( 1 + 53716725498338 T^{4} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 13041062 T^{2} + p^{10} T^{4} )^{4} \) |
| 31 | \( ( 1 + 6056 T + p^{5} T^{2} )^{8} \) |
| 37 | \( ( 1 - 9232649039873422 T^{4} + p^{20} T^{8} )^{2} \) |
| 41 | \( ( 1 - 152489362 T^{2} + p^{10} T^{4} )^{4} \) |
| 43 | \( ( 1 + 39483316792083506 T^{4} + p^{20} T^{8} )^{2} \) |
| 47 | \( ( 1 - 38535261982721662 T^{4} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 - 223599451091958862 T^{4} + p^{20} T^{8} )^{2} \) |
| 59 | \( ( 1 - 158955242 T^{2} + p^{10} T^{4} )^{4} \) |
| 61 | \( ( 1 - 29234 T + p^{5} T^{2} )^{8} \) |
| 67 | \( ( 1 + 2963513604903050066 T^{4} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1925755342 T^{2} + p^{10} T^{4} )^{4} \) |
| 73 | \( ( 1 - 7822328710261861342 T^{4} + p^{20} T^{8} )^{2} \) |
| 79 | \( ( 1 - 5077222942 T^{2} + p^{10} T^{4} )^{4} \) |
| 83 | \( ( 1 + 19466793867600919058 T^{4} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 + 7975834738 T^{2} + p^{10} T^{4} )^{4} \) |
| 97 | \( ( 1 + \)\(14\!\cdots\!06\)\( T^{4} + p^{20} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.18719104364055208858939577685, −3.91684985347544053066146322815, −3.90045440250568251757859254191, −3.89894279788623140931461106387, −3.83930533141775250311625810693, −3.58790546588771679154769080241, −3.48772928009142499769498360603, −3.39609102782407344892639883342, −2.94103560999191623162316705480, −2.88476502288304943563783311162, −2.75023229148040112488329988467, −2.70890068240126197794107443182, −2.18287726583529408962962683661, −2.06093608412952048825127394930, −1.89473903634621099802236650694, −1.85342735949705921874905385730, −1.76776433589307493872886339754, −1.73010463405311023789490232760, −1.50250947623069039364927151106, −0.944156765432997415592068258617, −0.75689046141659568286122862151, −0.56918887323220978029814816088, −0.50794773993415652033815420012, −0.47680490224857438205336748699, −0.19990642067150879162790229025,
0.19990642067150879162790229025, 0.47680490224857438205336748699, 0.50794773993415652033815420012, 0.56918887323220978029814816088, 0.75689046141659568286122862151, 0.944156765432997415592068258617, 1.50250947623069039364927151106, 1.73010463405311023789490232760, 1.76776433589307493872886339754, 1.85342735949705921874905385730, 1.89473903634621099802236650694, 2.06093608412952048825127394930, 2.18287726583529408962962683661, 2.70890068240126197794107443182, 2.75023229148040112488329988467, 2.88476502288304943563783311162, 2.94103560999191623162316705480, 3.39609102782407344892639883342, 3.48772928009142499769498360603, 3.58790546588771679154769080241, 3.83930533141775250311625810693, 3.89894279788623140931461106387, 3.90045440250568251757859254191, 3.91684985347544053066146322815, 4.18719104364055208858939577685
Plot not available for L-functions of degree greater than 10.
