| L(s) = 1 | + 8·11-s + 16-s − 4·81-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-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 + ⋯ |
| L(s) = 1 | + 8·11-s + 16-s − 4·81-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.646406959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646406959\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + T^{4} )^{2} \) |
| good | 3 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} )^{8} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 41 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{2} )^{8} \) |
| 61 | \( ( 1 + T^{2} )^{8} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{2} )^{8} \) |
| 97 | \( ( 1 + T^{4} )^{4} \) |
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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
−4.16808792919538325138385617295, −4.16095688480089205738650586775, −4.14826720011724330530796871942, −3.88068439778240306830719214553, −3.73635276484342410361127850386, −3.65931690518395841056680973446, −3.59180547247568932918178409059, −3.57301485632991461757942915631, −3.42009768972438794544522950462, −3.21234037999579597618764266069, −3.14911021694096398364553374199, −2.97566397800435230888191232088, −2.73043743462125967300242324941, −2.51535798108737488294081546008, −2.35929393547405430867988863662, −2.35406280460612877117047022117, −2.09774462166278239631058458330, −1.78771350263553708087260607097, −1.60644010234487088063850946054, −1.56004019184598016404971981003, −1.47228903260477238735748361695, −1.17540663682015430708949277025, −1.15519956538534857666304178869, −1.06478779223627187350958146069, −0.965022928397210697938829861752,
0.965022928397210697938829861752, 1.06478779223627187350958146069, 1.15519956538534857666304178869, 1.17540663682015430708949277025, 1.47228903260477238735748361695, 1.56004019184598016404971981003, 1.60644010234487088063850946054, 1.78771350263553708087260607097, 2.09774462166278239631058458330, 2.35406280460612877117047022117, 2.35929393547405430867988863662, 2.51535798108737488294081546008, 2.73043743462125967300242324941, 2.97566397800435230888191232088, 3.14911021694096398364553374199, 3.21234037999579597618764266069, 3.42009768972438794544522950462, 3.57301485632991461757942915631, 3.59180547247568932918178409059, 3.65931690518395841056680973446, 3.73635276484342410361127850386, 3.88068439778240306830719214553, 4.14826720011724330530796871942, 4.16095688480089205738650586775, 4.16808792919538325138385617295
Plot not available for L-functions of degree greater than 10.
