| L(s) = 1 | − 4·2-s + 4·4-s + 26·7-s + 16·8-s + 13·9-s − 104·14-s − 64·16-s − 48·17-s − 52·18-s + 74·23-s − 5·25-s + 104·28-s − 84·31-s + 64·32-s + 192·34-s + 52·36-s − 296·46-s − 288·47-s + 409·49-s + 20·50-s + 416·56-s + 336·62-s + 338·63-s + 192·64-s − 192·68-s + 424·71-s + 208·72-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 26/7·7-s + 2·8-s + 13/9·9-s − 7.42·14-s − 4·16-s − 2.82·17-s − 2.88·18-s + 3.21·23-s − 1/5·25-s + 26/7·28-s − 2.70·31-s + 2·32-s + 5.64·34-s + 13/9·36-s − 6.43·46-s − 6.12·47-s + 8.34·49-s + 2/5·50-s + 52/7·56-s + 5.41·62-s + 5.36·63-s + 3·64-s − 2.82·68-s + 5.97·71-s + 26/9·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.254403251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254403251\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - 13 T^{2} + 88 T^{4} - 13 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 14637 T^{4} + 2 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 24 T + 481 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 2 T^{2} - 130317 T^{4} - 2 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T + 840 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1667 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T + 1549 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 2198 T^{2} + 2957043 T^{4} + 2198 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 2039 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1717 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 144 T + 9121 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 1642 T^{2} - 5194317 T^{4} - 1642 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 6782 T^{2} + 33878163 T^{4} - 6782 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 7397 T^{2} + 40869768 T^{4} - 7397 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2^3$ | \( 1 + 1043 T^{2} - 19063272 T^{4} + 1043 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 106 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 30 T + 5629 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 64 T - 2145 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13373 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 27 T + 8164 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 1354 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.405731296927276691924548925431, −8.271999991869952717498950013110, −8.082211728825141523136275730305, −7.73944876521378333284289962428, −7.48388123781930313598676700890, −7.29193018079947758931373368298, −6.92089714510481752265309124119, −6.82065682675638026622331613184, −6.76145708743492918045462467420, −6.15987395720989264986071782856, −5.43237871643669336264016297007, −5.20859782621611941095977199987, −4.97399214704049161279017554339, −4.77619321466509051440040189030, −4.70820866465593003274126403506, −4.51396126080200381793766904377, −3.95189363526461196262778051074, −3.76082838458522684785980085941, −3.20281379555073375975421704880, −2.21576192428907453645310100360, −1.97695832520564097152588961337, −1.91027450398192168228517818785, −1.28468052418278069288704789907, −1.27465790756013093643970825717, −0.42655072750483009182435756357,
0.42655072750483009182435756357, 1.27465790756013093643970825717, 1.28468052418278069288704789907, 1.91027450398192168228517818785, 1.97695832520564097152588961337, 2.21576192428907453645310100360, 3.20281379555073375975421704880, 3.76082838458522684785980085941, 3.95189363526461196262778051074, 4.51396126080200381793766904377, 4.70820866465593003274126403506, 4.77619321466509051440040189030, 4.97399214704049161279017554339, 5.20859782621611941095977199987, 5.43237871643669336264016297007, 6.15987395720989264986071782856, 6.76145708743492918045462467420, 6.82065682675638026622331613184, 6.92089714510481752265309124119, 7.29193018079947758931373368298, 7.48388123781930313598676700890, 7.73944876521378333284289962428, 8.082211728825141523136275730305, 8.271999991869952717498950013110, 8.405731296927276691924548925431
