| L(s) = 1 | − 24·41-s + 64·61-s + 34·81-s − 96·101-s + 28·121-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 + ⋯ |
| L(s) = 1 | − 3.74·41-s + 8.19·61-s + 34/9·81-s − 9.55·101-s + 2.54·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3433176152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3433176152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - 17 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 142 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 383 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 3 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 1058 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 1057 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 3503 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12143 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−3.81115659019223745677181382222, −3.81057728120043960247904399083, −3.77663790450158198856260961574, −3.76443509031959078104109978565, −3.48311580131563074626154538058, −3.35553127923778365865551848490, −3.22492444896845732072199703959, −3.09742721915056338457548814040, −2.88795497183172696000309646231, −2.81049377305404378584507015731, −2.80985903649396970181983383971, −2.54407056403738125791462941562, −2.23774589387717210784102468451, −2.19079753897862136991284004612, −2.08956810449491771749331902660, −2.08775041315623824234372063847, −1.82708363683443197846961786982, −1.82137606757472024392519464420, −1.27896203491554908399397090144, −1.22936339631788947635942542825, −1.11428611765311907974525158793, −1.06452617722307528760107957366, −0.60932393319578629990833750474, −0.49562332881633218258206442361, −0.05947282740392097833968092392,
0.05947282740392097833968092392, 0.49562332881633218258206442361, 0.60932393319578629990833750474, 1.06452617722307528760107957366, 1.11428611765311907974525158793, 1.22936339631788947635942542825, 1.27896203491554908399397090144, 1.82137606757472024392519464420, 1.82708363683443197846961786982, 2.08775041315623824234372063847, 2.08956810449491771749331902660, 2.19079753897862136991284004612, 2.23774589387717210784102468451, 2.54407056403738125791462941562, 2.80985903649396970181983383971, 2.81049377305404378584507015731, 2.88795497183172696000309646231, 3.09742721915056338457548814040, 3.22492444896845732072199703959, 3.35553127923778365865551848490, 3.48311580131563074626154538058, 3.76443509031959078104109978565, 3.77663790450158198856260961574, 3.81057728120043960247904399083, 3.81115659019223745677181382222
Plot not available for L-functions of degree greater than 10.
