| L(s) = 1 | − 4·4-s + 8·7-s − 64·13-s + 12·16-s + 56·19-s + 64·25-s − 32·28-s + 32·31-s − 76·37-s − 88·43-s + 60·49-s + 256·52-s − 52·61-s − 32·64-s − 40·67-s + 224·73-s − 224·76-s + 104·79-s − 512·91-s + 32·97-s − 256·100-s − 112·103-s − 64·109-s + 96·112-s + 52·121-s − 128·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s + 8/7·7-s − 4.92·13-s + 3/4·16-s + 2.94·19-s + 2.55·25-s − 8/7·28-s + 1.03·31-s − 2.05·37-s − 2.04·43-s + 1.22·49-s + 4.92·52-s − 0.852·61-s − 1/2·64-s − 0.597·67-s + 3.06·73-s − 2.94·76-s + 1.31·79-s − 5.62·91-s + 0.329·97-s − 2.55·100-s − 1.08·103-s − 0.587·109-s + 6/7·112-s + 0.429·121-s − 1.03·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.312110714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312110714\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2031 T^{4} - 64 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T - 6 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 32 T + 567 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 31551 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 28 T + 810 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1312 T^{2} + 1790223 T^{4} - 1312 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 38 T + 1371 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4960 T^{2} + 11739714 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 44 T + 3210 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4130 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3424 T^{2} + 8198754 T^{4} - 3424 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1828 T^{2} + 20031270 T^{4} - 1828 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 + 20 T + 8106 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9652 T^{2} + 49230438 T^{4} - 9652 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 112 T + 13551 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 52 T + 7866 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16612 T^{2} + 136472550 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 7456 T^{2} + 104845119 T^{4} - 7456 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 11970 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.280163248393608496049934331649, −9.264583324407458929940317232518, −8.411301001047979272813089975474, −8.409774595413748179119831159457, −8.218352165724532609751907934433, −7.73411137293345524284399354022, −7.39830559953312551651555052357, −7.38556907458689586132010136765, −7.09921080175870541973625491972, −6.74574591659903942178735779389, −6.64288782676863942523476843447, −5.80429959425865506922202967372, −5.37867930653519245751032729566, −5.13477280980763541948708370669, −5.04955060995645206522782137467, −4.89008171240611499971508953300, −4.65771532530577310090220744271, −4.31801430535720735154613493551, −3.44886686862399144321536145439, −3.24597550965459115830460010404, −2.86487576909049947862593703507, −2.39784246845985934221107743062, −1.94274724425576045271595004294, −1.13041756349053652121415970419, −0.44371823779456661250118975195,
0.44371823779456661250118975195, 1.13041756349053652121415970419, 1.94274724425576045271595004294, 2.39784246845985934221107743062, 2.86487576909049947862593703507, 3.24597550965459115830460010404, 3.44886686862399144321536145439, 4.31801430535720735154613493551, 4.65771532530577310090220744271, 4.89008171240611499971508953300, 5.04955060995645206522782137467, 5.13477280980763541948708370669, 5.37867930653519245751032729566, 5.80429959425865506922202967372, 6.64288782676863942523476843447, 6.74574591659903942178735779389, 7.09921080175870541973625491972, 7.38556907458689586132010136765, 7.39830559953312551651555052357, 7.73411137293345524284399354022, 8.218352165724532609751907934433, 8.409774595413748179119831159457, 8.411301001047979272813089975474, 9.264583324407458929940317232518, 9.280163248393608496049934331649
