L(s) = 1 | + 16·13-s − 8·37-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
L(s) = 1 | + 16·13-s − 8·37-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(20.13026573\) |
\(L(\frac12)\) |
\(\approx\) |
\(20.13026573\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{8} + T^{16} \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T^{8} + T^{16} \) |
good | 3 | \( 1 - T^{16} + T^{32} \) |
| 5 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
| 11 | \( 1 - T^{16} + T^{32} \) |
| 13 | \( ( 1 - T )^{16}( 1 + T^{2} )^{8} \) |
| 19 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 23 | \( 1 - T^{16} + T^{32} \) |
| 29 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 31 | \( 1 - T^{16} + T^{32} \) |
| 37 | \( ( 1 + T + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \) |
| 41 | \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{8} )^{4} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 53 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \) |
| 67 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 71 | \( ( 1 + T^{16} )^{2} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \) |
| 79 | \( 1 - T^{16} + T^{32} \) |
| 83 | \( ( 1 + T^{8} )^{4} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 97 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
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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
−2.24927129027896035769854519641, −2.18350233247883020628171431148, −2.13742003438485068584642975382, −2.12197876064999204156885235052, −1.95582117272900423141083041928, −1.84418935269406588962323084411, −1.79750003065891144287353001632, −1.72176382044842997479479630411, −1.70371929839305421147873953388, −1.63391900950788641680776299569, −1.53618531007997584947266727255, −1.50758829616401443202133940713, −1.49894270594565298247226418717, −1.41372263364166987020435994159, −1.39845448282911060595235905227, −1.32329369495155567336578317040, −1.14670767376512373667812128755, −1.14387686549834291822959227425, −1.06866995795516995970129382761, −0.981723800371985944790888220082, −0.965952040364417253841946403619, −0.875764897751903841464627305327, −0.66786727119881126711942151945, −0.51903379653606829551190515665, −0.50519564394907193921595748347,
0.50519564394907193921595748347, 0.51903379653606829551190515665, 0.66786727119881126711942151945, 0.875764897751903841464627305327, 0.965952040364417253841946403619, 0.981723800371985944790888220082, 1.06866995795516995970129382761, 1.14387686549834291822959227425, 1.14670767376512373667812128755, 1.32329369495155567336578317040, 1.39845448282911060595235905227, 1.41372263364166987020435994159, 1.49894270594565298247226418717, 1.50758829616401443202133940713, 1.53618531007997584947266727255, 1.63391900950788641680776299569, 1.70371929839305421147873953388, 1.72176382044842997479479630411, 1.79750003065891144287353001632, 1.84418935269406588962323084411, 1.95582117272900423141083041928, 2.12197876064999204156885235052, 2.13742003438485068584642975382, 2.18350233247883020628171431148, 2.24927129027896035769854519641
Plot not available for L-functions of degree greater than 10.