| L(s) = 1 | − 5-s − 9-s + 2·13-s + 3·37-s + 2·41-s + 45-s − 4·49-s − 3·53-s − 2·65-s − 3·89-s + 5·113-s − 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s − 3·185-s + ⋯ |
| L(s) = 1 | − 5-s − 9-s + 2·13-s + 3·37-s + 2·41-s + 45-s − 4·49-s − 3·53-s − 2·65-s − 3·89-s + 5·113-s − 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s − 3·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.039006463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039006463\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.23838537211485921702912498203, −6.14458728011958769585507479192, −6.10765773794157223890110576121, −5.75377103737526064573708768654, −5.70150332984972692526655503962, −5.27383778758679779649587760825, −5.04172528733678834429827788973, −4.96704805545548452962532032260, −4.58167232460276288284455352005, −4.50680658835662007852365179417, −4.15855858873305867655114064649, −4.15702597334876755151653822199, −3.91920670958254897482281886614, −3.61135086686313717219857214790, −3.21046568866023685678916216702, −3.16348663325660160583943718081, −3.13350887587867783382647697112, −2.70439904790741946822667987250, −2.65872644772428954045324928006, −2.11710160794045514991410763676, −1.77613831638991402525053912138, −1.65884713252355986403527054651, −1.26244889228972097697139828913, −0.926360685319392811100748168732, −0.47057475450149827222893662640,
0.47057475450149827222893662640, 0.926360685319392811100748168732, 1.26244889228972097697139828913, 1.65884713252355986403527054651, 1.77613831638991402525053912138, 2.11710160794045514991410763676, 2.65872644772428954045324928006, 2.70439904790741946822667987250, 3.13350887587867783382647697112, 3.16348663325660160583943718081, 3.21046568866023685678916216702, 3.61135086686313717219857214790, 3.91920670958254897482281886614, 4.15702597334876755151653822199, 4.15855858873305867655114064649, 4.50680658835662007852365179417, 4.58167232460276288284455352005, 4.96704805545548452962532032260, 5.04172528733678834429827788973, 5.27383778758679779649587760825, 5.70150332984972692526655503962, 5.75377103737526064573708768654, 6.10765773794157223890110576121, 6.14458728011958769585507479192, 6.23838537211485921702912498203
