| L(s) = 1 | − 4·7-s + 24·17-s + 12·23-s + 24·41-s + 12·47-s + 8·49-s + 32·73-s − 96·79-s − 8·81-s − 32·97-s + 28·103-s − 96·119-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 5.82·17-s + 2.50·23-s + 3.74·41-s + 1.75·47-s + 8/7·49-s + 3.74·73-s − 10.8·79-s − 8/9·81-s − 3.24·97-s + 2.75·103-s − 8.80·119-s + 4.36·121-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 − 3.78·161-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933754034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933754034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 8 T^{4} + 94 T^{8} + 8 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 82 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 6 T + 18 T^{2} - 102 T^{3} + 542 T^{4} - 102 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 64 T^{2} + 2190 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2338 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 2632 T^{4} + 3107358 T^{8} + 2632 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 6 T + 18 T^{2} - 246 T^{3} + 3326 T^{4} - 246 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2846 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 112 T^{2} + 9822 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 56 T^{4} - 39549474 T^{8} - 56 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 80 T^{2} + 4878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 7838 T^{4} - 1008 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 8200 T^{4} + 98353758 T^{8} + 8200 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 116 T^{2} + 7110 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 15038 T^{4} + 1392 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.06976475420929199685616595702, −3.96169076421326649443595374970, −3.82964992408798899629724745625, −3.81458870462094161053880747338, −3.64973315563314890300865915358, −3.36559306393131825937531644653, −3.18301493576272530702818587033, −3.15170224225844111490111025225, −3.12672718798692322655040835809, −2.89419499199611230187226299300, −2.88900741875531642491218896892, −2.71555211052575379549109709256, −2.69029636376028250751587944539, −2.52538297335098920023651730146, −2.27759991576514638845229845020, −2.11995580734854368621407193244, −1.71644503586669524456608904952, −1.61747445952839650692660786368, −1.46695915652241174708754803727, −1.16776725484021467645366660162, −1.11590175690007934083829232244, −1.03865617724658463972009393088, −0.77721436915355036961649964790, −0.72526471791695536047133783061, −0.12196649610526455167181412303,
0.12196649610526455167181412303, 0.72526471791695536047133783061, 0.77721436915355036961649964790, 1.03865617724658463972009393088, 1.11590175690007934083829232244, 1.16776725484021467645366660162, 1.46695915652241174708754803727, 1.61747445952839650692660786368, 1.71644503586669524456608904952, 2.11995580734854368621407193244, 2.27759991576514638845229845020, 2.52538297335098920023651730146, 2.69029636376028250751587944539, 2.71555211052575379549109709256, 2.88900741875531642491218896892, 2.89419499199611230187226299300, 3.12672718798692322655040835809, 3.15170224225844111490111025225, 3.18301493576272530702818587033, 3.36559306393131825937531644653, 3.64973315563314890300865915358, 3.81458870462094161053880747338, 3.82964992408798899629724745625, 3.96169076421326649443595374970, 4.06976475420929199685616595702
Plot not available for L-functions of degree greater than 10.
