| L(s) = 1 | − 4·7-s + 16·16-s + 8·19-s + 44·31-s + 32·37-s + 8·49-s + 64·61-s − 4·67-s + 44·73-s + 16·79-s + 20·97-s − 28·109-s − 64·112-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4·16-s + 1.83·19-s + 7.90·31-s + 5.26·37-s + 8/7·49-s + 8.19·61-s − 0.488·67-s + 5.14·73-s + 1.80·79-s + 2.03·97-s − 2.68·109-s − 6.04·112-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(33.73899931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(33.73899931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} + 12 T^{3} + 71 T^{4} + 12 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 196 T^{4} + 23334 T^{8} - 196 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 20 T^{2} + 246 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T + 8 T^{2} + 12 T^{3} - 466 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 7262 T^{4} - 1008 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 2372 T^{4} + 3712710 T^{8} + 2372 p^{4} T^{12} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 94 T^{2} + 5475 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 476 T^{4} + 4588806 T^{8} + 476 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 100 T^{2} + 7686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 188 T^{4} - 22801242 T^{8} + 188 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 2 T + 2 T^{2} - 12 T^{3} - 5257 T^{4} - 12 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 9884 T^{4} + 56195526 T^{8} + 9884 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 22 T + 242 T^{2} - 2640 T^{3} + 26591 T^{4} - 2640 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 135 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 12068 T^{4} + 72822630 T^{8} + 12068 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 4732 T^{4} + 3238086 T^{8} - 4732 p^{4} T^{12} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 10 T + 50 T^{2} - 360 T^{3} - 1129 T^{4} - 360 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.90244046812965131845945148989, −3.89841279216720421223553617619, −3.86311366267665008219555066694, −3.82043736804219547946068688337, −3.43143511807349684337316015634, −3.36091382122942962873339077779, −3.32399784881215919016434283531, −2.92455715440164857312266655250, −2.86542942240841223908089264154, −2.80120782473260684373775638853, −2.74648931292868157520106337156, −2.73852132499186069970692801832, −2.62646316497885136245112809804, −2.32547896359035603632783019556, −2.14194742852933901132763786012, −2.07344892165225897478391088281, −2.06801622710132987630697170484, −1.43163311641835119154542260423, −1.15999620541716509021433302956, −1.05874145512499097715464359748, −1.03624748412414464447182574232, −0.976731997049602755555705865548, −0.819455074161333176705525559634, −0.60743554939368986921247023122, −0.60062570439484950742705734875,
0.60062570439484950742705734875, 0.60743554939368986921247023122, 0.819455074161333176705525559634, 0.976731997049602755555705865548, 1.03624748412414464447182574232, 1.05874145512499097715464359748, 1.15999620541716509021433302956, 1.43163311641835119154542260423, 2.06801622710132987630697170484, 2.07344892165225897478391088281, 2.14194742852933901132763786012, 2.32547896359035603632783019556, 2.62646316497885136245112809804, 2.73852132499186069970692801832, 2.74648931292868157520106337156, 2.80120782473260684373775638853, 2.86542942240841223908089264154, 2.92455715440164857312266655250, 3.32399784881215919016434283531, 3.36091382122942962873339077779, 3.43143511807349684337316015634, 3.82043736804219547946068688337, 3.86311366267665008219555066694, 3.89841279216720421223553617619, 3.90244046812965131845945148989
Plot not available for L-functions of degree greater than 10.
