| L(s) = 1 | − 4·4-s − 3·5-s + 5·9-s + 4·16-s + 12·20-s + 5·25-s − 10·31-s − 20·36-s − 15·45-s − 14·49-s + 30·59-s + 6·71-s − 12·80-s + 9·81-s + 72·89-s − 20·100-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + 149-s + 151-s + 30·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.34·5-s + 5/3·9-s + 16-s + 2.68·20-s + 25-s − 1.79·31-s − 3.33·36-s − 2.23·45-s − 2·49-s + 3.90·59-s + 0.712·71-s − 1.34·80-s + 81-s + 7.63·89-s − 2·100-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 2.40·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7729854498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7729854498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 3 T + 4 T^{2} - 3 T^{3} - 29 T^{4} - 3 p T^{5} + 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 3 | \( ( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 5 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 9 T + p T^{2} )^{4}( 1 + 9 T + p T^{2} )^{4} \) |
| 29 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 5 T - 6 T^{2} - 185 T^{3} - 739 T^{4} - 185 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 7 T + 12 T^{2} + 175 T^{3} - 1669 T^{4} + 175 p T^{5} + 12 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )( 1 + 7 T + 12 T^{2} - 175 T^{3} - 1669 T^{4} - 175 p T^{5} + 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 600 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 53 | \( ( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 1605 p T^{5} + 166 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 13 T + p T^{2} )^{4}( 1 + 13 T + p T^{2} )^{4} \) |
| 71 | \( ( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 1615 p T^{5} + 192 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )( 1 + 17 T + 192 T^{2} + 1615 T^{3} + 8831 T^{4} + 1615 p T^{5} + 192 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 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
−4.69264408631960314296060749862, −4.61156897956159086415183557603, −4.30552289438127485679125682674, −4.28419428340696965942116992930, −4.06803189888003123442656227330, −4.02340587917600162449237868880, −3.92264737607168538797286251096, −3.57362900969374760079999479711, −3.50886403207379151971609900768, −3.43983452334856242028598547134, −3.34260277533087288524714138451, −3.32447350358020980501291274959, −3.24053637709502913762680422962, −2.55058556294639325246723229561, −2.47666909111751719761806772299, −2.37997738864845601231938921870, −2.22543299750200485659716120009, −2.10549111049262770821985230918, −1.90318659873218789095566826278, −1.51759196513400767234016936466, −1.28244606398199249642868335926, −1.06239790044189189762405681112, −0.967322388999460005605743076272, −0.42203143988934147925154091636, −0.26996600427650014263258106451,
0.26996600427650014263258106451, 0.42203143988934147925154091636, 0.967322388999460005605743076272, 1.06239790044189189762405681112, 1.28244606398199249642868335926, 1.51759196513400767234016936466, 1.90318659873218789095566826278, 2.10549111049262770821985230918, 2.22543299750200485659716120009, 2.37997738864845601231938921870, 2.47666909111751719761806772299, 2.55058556294639325246723229561, 3.24053637709502913762680422962, 3.32447350358020980501291274959, 3.34260277533087288524714138451, 3.43983452334856242028598547134, 3.50886403207379151971609900768, 3.57362900969374760079999479711, 3.92264737607168538797286251096, 4.02340587917600162449237868880, 4.06803189888003123442656227330, 4.28419428340696965942116992930, 4.30552289438127485679125682674, 4.61156897956159086415183557603, 4.69264408631960314296060749862
Plot not available for L-functions of degree greater than 10.
