| L(s) = 1 | − 8·5-s + 6·9-s − 2·25-s − 32·29-s + 200·41-s − 48·45-s − 172·49-s − 104·61-s + 27·81-s − 344·89-s − 128·101-s + 520·109-s − 20·121-s + 344·125-s + 127-s + 131-s + 137-s + 139-s + 256·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 508·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 2/3·9-s − 0.0799·25-s − 1.10·29-s + 4.87·41-s − 1.06·45-s − 3.51·49-s − 1.70·61-s + 1/3·81-s − 3.86·89-s − 1.26·101-s + 4.77·109-s − 0.165·121-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.76·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.00·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9181302637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9181302637\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 254 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 494 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 638 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2066 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4862 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6710 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 5906 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 5806 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 94 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 3550 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 6722 T^{2} + p^{4} T^{4} )^{2} \) |
| 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
−8.601960275856539373127998428097, −8.216481235622918693462660942337, −7.994127565974502589640118152274, −7.81490521542336588913280168592, −7.49246109314488562746666276733, −7.46416596975878565569260182820, −7.22125452367053712072624122484, −6.83716122833328569476025853666, −6.42307191133223129071561502552, −6.07130266966711418854556789538, −6.05610197200564358484132590882, −5.64009371386387508719230951587, −5.29043133128630150328629315793, −4.84132006420214184274432350303, −4.52755326457034555390583545300, −4.31335061594450515379549059910, −3.95676197980158678407653602038, −3.92551232536884664585227996651, −3.36126982935995182877524613785, −2.88173306175734402365578345380, −2.81529821858172308101933299975, −1.95146123508546104146320314791, −1.70813606476582332066516456832, −0.973815696874001550623032216805, −0.30179231113352374400280791126,
0.30179231113352374400280791126, 0.973815696874001550623032216805, 1.70813606476582332066516456832, 1.95146123508546104146320314791, 2.81529821858172308101933299975, 2.88173306175734402365578345380, 3.36126982935995182877524613785, 3.92551232536884664585227996651, 3.95676197980158678407653602038, 4.31335061594450515379549059910, 4.52755326457034555390583545300, 4.84132006420214184274432350303, 5.29043133128630150328629315793, 5.64009371386387508719230951587, 6.05610197200564358484132590882, 6.07130266966711418854556789538, 6.42307191133223129071561502552, 6.83716122833328569476025853666, 7.22125452367053712072624122484, 7.46416596975878565569260182820, 7.49246109314488562746666276733, 7.81490521542336588913280168592, 7.994127565974502589640118152274, 8.216481235622918693462660942337, 8.601960275856539373127998428097
