L(s) = 1 | + 4·4-s + 8·9-s + 10·16-s − 64·29-s + 32·36-s + 36·64-s − 4·81-s − 256·116-s + 136·121-s + 127-s + 131-s + 137-s + 139-s + 80·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 2·4-s + 8/3·9-s + 5/2·16-s − 11.8·29-s + 16/3·36-s + 9/2·64-s − 4/9·81-s − 23.7·116-s + 12.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.357776718\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.357776718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 30 T^{4} + p^{4} T^{8} )^{2} \) |
good | 3 | \( ( 1 - 2 T^{2} + 11 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 34 T^{2} + 523 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{8} \) |
| 17 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 19 | \( ( 1 + 18 T^{2} + 795 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 72 T^{2} + 2322 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{8} \) |
| 31 | \( ( 1 + 104 T^{2} + 4594 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 + 132 T^{2} + 7926 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 8 T^{2} + 1842 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 - 48 T^{2} + 1586 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 196 T^{2} + 16438 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{8} \) |
| 67 | \( ( 1 + 50 T^{2} + 6715 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 168 T^{2} + 17106 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 - 236 T^{2} + 25894 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 322 T^{2} + 39691 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{8} \) |
| 97 | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.90522380321206627581038274368, −2.68880666049150549261804401125, −2.54010869945049489556278359538, −2.36997082603067310370626892992, −2.30297527019478705368829473507, −2.25236513094011303920686302327, −2.22055793327888098324109646840, −2.07035189837742040366633173750, −2.04626337067381079835487379965, −1.97330839285529878568808295013, −1.86250211906276519904125108829, −1.84838887714029834901894860392, −1.77881900216931862337889865240, −1.75745661457461597012816519640, −1.73477651051829381818562220744, −1.57716017248780894564658079409, −1.35271129022819255011565257173, −1.26835568632576904493380444645, −1.17538802946832541639135451976, −1.00094795237487868752861951671, −0.958816106217446426112407318102, −0.63451537824080754610578586289, −0.55517751089352580839938519974, −0.29103987232644041766081589300, −0.11487241855082086255365377024,
0.11487241855082086255365377024, 0.29103987232644041766081589300, 0.55517751089352580839938519974, 0.63451537824080754610578586289, 0.958816106217446426112407318102, 1.00094795237487868752861951671, 1.17538802946832541639135451976, 1.26835568632576904493380444645, 1.35271129022819255011565257173, 1.57716017248780894564658079409, 1.73477651051829381818562220744, 1.75745661457461597012816519640, 1.77881900216931862337889865240, 1.84838887714029834901894860392, 1.86250211906276519904125108829, 1.97330839285529878568808295013, 2.04626337067381079835487379965, 2.07035189837742040366633173750, 2.22055793327888098324109646840, 2.25236513094011303920686302327, 2.30297527019478705368829473507, 2.36997082603067310370626892992, 2.54010869945049489556278359538, 2.68880666049150549261804401125, 2.90522380321206627581038274368
Plot not available for L-functions of degree greater than 10.