| L(s) = 1 | + 4·4-s − 8·13-s + 7·16-s + 32·25-s + 16·37-s + 8·49-s − 32·52-s − 32·61-s + 8·64-s − 32·73-s − 32·97-s + 128·100-s − 8·109-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.21·13-s + 7/4·16-s + 32/5·25-s + 2.63·37-s + 8/7·49-s − 4.43·52-s − 4.09·61-s + 64-s − 3.74·73-s − 3.24·97-s + 64/5·100-s − 0.766·109-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.253016033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.253016033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 16 T^{2} + 111 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T^{2} + 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 8 T^{2} + 6 p T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 40 T^{2} + 903 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T^{2} + 618 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 32 T^{2} + 546 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 88 T^{2} + 3543 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 52 T^{2} + 1830 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 75 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 88 T^{2} + 4530 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 148 T^{2} + 9162 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 140 T^{2} + 9270 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 188 T^{2} + 15750 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 244 T^{2} + 23850 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 176 T^{2} + 16098 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 196 T^{2} + 18618 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 188 T^{2} + 19542 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 160 T^{2} + 15039 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 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
−5.02339689307846483482405348472, −4.96450300340809648173942071589, −4.89886337943920601482846232778, −4.76305405798207977857441001278, −4.68798976197116905751440568680, −4.66842547333559660448373565010, −4.29182870042305271009160013761, −4.15170827723867320225682135473, −4.05996575920765194667503528875, −3.83056655101392189839674828186, −3.67116412474480569795861328867, −3.05882932880258372018172504584, −3.04192445635136276950692817894, −3.02139662498620363561143166853, −3.00689490415918127426876835234, −2.89733174672308342715945793647, −2.55143431722386095096380326296, −2.37696786619257676036627068993, −2.36832048792969856479523889828, −2.17603027309688124409642496843, −1.58304293381580849473353851010, −1.36381546283972589962844729878, −1.26526835187240520540772291507, −1.13378979768073545842853297106, −0.42863723013195417110561148250,
0.42863723013195417110561148250, 1.13378979768073545842853297106, 1.26526835187240520540772291507, 1.36381546283972589962844729878, 1.58304293381580849473353851010, 2.17603027309688124409642496843, 2.36832048792969856479523889828, 2.37696786619257676036627068993, 2.55143431722386095096380326296, 2.89733174672308342715945793647, 3.00689490415918127426876835234, 3.02139662498620363561143166853, 3.04192445635136276950692817894, 3.05882932880258372018172504584, 3.67116412474480569795861328867, 3.83056655101392189839674828186, 4.05996575920765194667503528875, 4.15170827723867320225682135473, 4.29182870042305271009160013761, 4.66842547333559660448373565010, 4.68798976197116905751440568680, 4.76305405798207977857441001278, 4.89886337943920601482846232778, 4.96450300340809648173942071589, 5.02339689307846483482405348472
Plot not available for L-functions of degree greater than 10.
