| L(s) = 1 | + 4-s − 4·9-s + 12·11-s − 4·16-s − 16·19-s − 12·29-s + 8·31-s − 4·36-s + 48·41-s + 12·44-s + 8·49-s − 4·59-s + 20·61-s + 8·64-s − 24·71-s − 16·76-s + 40·79-s + 10·81-s − 56·89-s − 48·99-s + 32·101-s + 4·109-s − 12·116-s + 44·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4/3·9-s + 3.61·11-s − 16-s − 3.67·19-s − 2.22·29-s + 1.43·31-s − 2/3·36-s + 7.49·41-s + 1.80·44-s + 8/7·49-s − 0.520·59-s + 2.56·61-s + 64-s − 2.84·71-s − 1.83·76-s + 4.50·79-s + 10/9·81-s − 5.93·89-s − 4.82·99-s + 3.18·101-s + 0.383·109-s − 1.11·116-s + 4·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.769680355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.769680355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - T^{2} + 5 T^{4} - 17 T^{6} + 13 T^{8} - 17 p^{2} T^{10} + 5 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 8 T^{2} + 92 T^{4} - 376 T^{6} + 3334 T^{8} - 376 p^{2} T^{10} + 92 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 6 T + 32 T^{2} - 10 p T^{3} + 446 T^{4} - 10 p^{2} T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 24 T^{2} + 380 T^{4} - 2728 T^{6} + 21798 T^{8} - 2728 p^{2} T^{10} + 380 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 45 T^{2} + 1073 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 8 T + 77 T^{2} + 396 T^{3} + 2229 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 82 T^{2} + 3227 T^{4} - 83644 T^{6} + 1872189 T^{8} - 83644 p^{2} T^{10} + 3227 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T + 80 T^{2} + 482 T^{3} + 3038 T^{4} + 482 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 73 T^{2} - 452 T^{3} + 2629 T^{4} - 452 p T^{5} + 73 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 108 T^{2} + 9412 T^{4} - 496756 T^{6} + 22182614 T^{8} - 496756 p^{2} T^{10} + 9412 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 24 T + 348 T^{2} - 3392 T^{3} + 25094 T^{4} - 3392 p T^{5} + 348 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 100 T^{2} + 8756 T^{4} - 512700 T^{6} + 25722486 T^{8} - 512700 p^{2} T^{10} + 8756 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 165 T^{2} + 11193 T^{4} - 165 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 242 T^{2} + 32267 T^{4} - 2818524 T^{6} + 175851069 T^{8} - 2818524 p^{2} T^{10} + 32267 p^{4} T^{12} - 242 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 2 T + 92 T^{2} + 34 T^{3} + 6214 T^{4} + 34 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T + 191 T^{2} - 1360 T^{3} + 16641 T^{4} - 1360 p T^{5} + 191 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 280 T^{2} + 44636 T^{4} - 4824360 T^{6} + 375137766 T^{8} - 4824360 p^{2} T^{10} + 44636 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 12 T + 188 T^{2} + 2164 T^{3} + 17270 T^{4} + 2164 p T^{5} + 188 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 532 T^{2} + 127172 T^{4} - 17839404 T^{6} + 1606277814 T^{8} - 17839404 p^{2} T^{10} + 127172 p^{4} T^{12} - 532 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 20 T + 341 T^{2} - 4240 T^{3} + 42121 T^{4} - 4240 p T^{5} + 341 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 394 T^{2} + 78395 T^{4} - 10278788 T^{6} + 984147973 T^{8} - 10278788 p^{2} T^{10} + 78395 p^{4} T^{12} - 394 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 28 T + 552 T^{2} + 7316 T^{3} + 80094 T^{4} + 7316 p T^{5} + 552 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 220 T^{2} + 30436 T^{4} - 2882340 T^{6} + 293829686 T^{8} - 2882340 p^{2} T^{10} + 30436 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−5.08734384044059536173282204868, −4.78550520274607635047902103806, −4.67173347824719704141828490813, −4.39700464961132007042730586452, −4.31124514978421649190338603238, −4.25814270025030682342207605766, −4.17134390110150705919403294782, −4.10889232007677163108009681092, −4.10418969661550476310475732969, −3.75673191286565757675778590039, −3.59290800591646553697813162213, −3.39519526275468212979743315100, −3.34486114651084063591358058358, −2.85832935609229938973881929006, −2.73693626911397138241946742233, −2.65729844375092725379041936140, −2.49324380308616866714922418345, −2.23648002572703018875253857740, −2.11858504209262853722309557137, −1.94346380612164434724153006046, −1.73689271650012716462459214339, −1.49313346082978620012681099129, −0.964328604534761658639698097555, −0.68616627219879957663938109550, −0.63511139415157231085171383201,
0.63511139415157231085171383201, 0.68616627219879957663938109550, 0.964328604534761658639698097555, 1.49313346082978620012681099129, 1.73689271650012716462459214339, 1.94346380612164434724153006046, 2.11858504209262853722309557137, 2.23648002572703018875253857740, 2.49324380308616866714922418345, 2.65729844375092725379041936140, 2.73693626911397138241946742233, 2.85832935609229938973881929006, 3.34486114651084063591358058358, 3.39519526275468212979743315100, 3.59290800591646553697813162213, 3.75673191286565757675778590039, 4.10418969661550476310475732969, 4.10889232007677163108009681092, 4.17134390110150705919403294782, 4.25814270025030682342207605766, 4.31124514978421649190338603238, 4.39700464961132007042730586452, 4.67173347824719704141828490813, 4.78550520274607635047902103806, 5.08734384044059536173282204868
Plot not available for L-functions of degree greater than 10.
