| L(s) = 1 | + 6·3-s − 5-s + 15·9-s + 3·11-s − 6·15-s − 9·17-s + 19-s − 12·23-s + 5·25-s + 18·27-s − 2·29-s − 31-s + 18·33-s + 27·37-s + 30·41-s − 15·45-s + 15·47-s − 54·51-s − 3·53-s − 3·55-s + 6·57-s − 59-s − 12·61-s − 18·67-s − 72·69-s + 12·71-s + 15·73-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 0.447·5-s + 5·9-s + 0.904·11-s − 1.54·15-s − 2.18·17-s + 0.229·19-s − 2.50·23-s + 25-s + 3.46·27-s − 0.371·29-s − 0.179·31-s + 3.13·33-s + 4.43·37-s + 4.68·41-s − 2.23·45-s + 2.18·47-s − 7.56·51-s − 0.412·53-s − 0.404·55-s + 0.794·57-s − 0.130·59-s − 1.53·61-s − 2.19·67-s − 8.66·69-s + 1.42·71-s + 1.75·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.38092805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.38092805\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T + 63 T^{2} + 324 T^{3} + 1466 T^{4} + 324 p T^{5} + 63 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 14 T^{3} + 196 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 14 T^{3} + 1312 T^{4} - 14 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 27 T + 373 T^{2} - 3510 T^{3} + 24522 T^{4} - 3510 p T^{5} + 373 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 15 T + 183 T^{2} - 1620 T^{3} + 12980 T^{4} - 1620 p T^{5} + 183 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T + 67 T^{2} + 192 T^{3} + 1446 T^{4} + 192 p T^{5} + 67 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 43 T^{2} - 252 T^{3} - 2304 T^{4} - 252 p T^{5} + 43 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 193 T^{2} + 1530 T^{3} + 9972 T^{4} + 1530 p T^{5} + 193 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 15 T + 235 T^{2} - 2400 T^{3} + 25746 T^{4} - 2400 p T^{5} + 235 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 14 p T^{5} - 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62235158519014918214147507618, −7.03612924724030584450084137808, −6.83609233709056375197379581958, −6.38962675923068870423781790407, −6.29175966788611932923959960166, −6.15059512883423555085305806111, −6.04164448305357931036516488694, −5.53502414156249029738533042188, −5.48614917378650306179413362076, −4.75793072180921361689850446533, −4.58302768564089329542421197940, −4.49014893037245897602809841794, −4.01942505981022545473454502387, −3.98829940797767663379501254858, −3.87633730315401775075689056968, −3.73697213701862280626243310236, −3.06118209029767841100099964641, −2.74579446592009931401234472719, −2.66836745540452429146182002237, −2.61510398619315868829920328450, −2.38016032719902969222988574103, −1.94130971427855647793677836062, −1.69980048485353719383561222284, −0.945252816187647753011242282979, −0.63695643960673285640660315023,
0.63695643960673285640660315023, 0.945252816187647753011242282979, 1.69980048485353719383561222284, 1.94130971427855647793677836062, 2.38016032719902969222988574103, 2.61510398619315868829920328450, 2.66836745540452429146182002237, 2.74579446592009931401234472719, 3.06118209029767841100099964641, 3.73697213701862280626243310236, 3.87633730315401775075689056968, 3.98829940797767663379501254858, 4.01942505981022545473454502387, 4.49014893037245897602809841794, 4.58302768564089329542421197940, 4.75793072180921361689850446533, 5.48614917378650306179413362076, 5.53502414156249029738533042188, 6.04164448305357931036516488694, 6.15059512883423555085305806111, 6.29175966788611932923959960166, 6.38962675923068870423781790407, 6.83609233709056375197379581958, 7.03612924724030584450084137808, 7.62235158519014918214147507618
