| L(s) = 1 | + 8·7-s + 32·13-s + 32·19-s + 8·25-s + 8·31-s + 104·37-s + 112·43-s − 156·49-s + 120·61-s + 336·67-s − 112·73-s + 152·79-s + 256·91-s − 352·97-s + 344·103-s + 512·109-s − 12·121-s + 127-s + 131-s + 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 2.46·13-s + 1.68·19-s + 8/25·25-s + 8/31·31-s + 2.81·37-s + 2.60·43-s − 3.18·49-s + 1.96·61-s + 5.01·67-s − 1.53·73-s + 1.92·79-s + 2.81·91-s − 3.62·97-s + 3.33·103-s + 4.69·109-s − 0.0991·121-s + 0.00787·127-s + 0.00763·131-s + 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.51836513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.51836513\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 914 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 21370 T^{4} + 12 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 314 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 800 T^{2} + 325634 T^{4} - 800 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 434 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 564 T^{2} + 436454 T^{4} - 564 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2600 T^{2} + 3045074 T^{4} - 2600 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 518 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5888 T^{2} + 14148290 T^{4} - 5888 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 56 T + 3074 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 2309030 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8680 T^{2} + 33219474 T^{4} - 8680 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 60 T + 6934 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 168 T + 15682 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12596 T^{2} + 79584614 T^{4} - 12596 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 56 T - 1230 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 76 T + 12518 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24244 T^{2} + 241403334 T^{4} - 24244 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 10176 T^{2} + 39402434 T^{4} - 10176 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 176 T + 26210 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.79384640444496381011296952785, −6.46243375406920911220503645520, −6.30777428209475975266412238365, −6.18063988834624981017115662548, −6.03131209356966241246113251243, −5.52857228721488793390946439266, −5.46139802300402566936098390033, −5.31289648926282314230319992294, −5.14021515441890065114126707089, −4.51708024808750445496230570398, −4.48461239067124809059727420586, −4.46981627935296430970816564571, −3.97392469223757199181451429030, −3.82795662696126431535837649972, −3.34581264028070520781275344117, −3.25300036473252915884705249000, −3.21883290576343141449284361679, −2.61337794086476379827969693424, −2.35408784622799217644785882796, −1.92069351415047393407405641465, −1.90225405166918851832332336318, −1.24285799093348495760287214332, −0.975103988790914398675853047078, −0.834862565627627880500164392934, −0.60470690133757040221524977002,
0.60470690133757040221524977002, 0.834862565627627880500164392934, 0.975103988790914398675853047078, 1.24285799093348495760287214332, 1.90225405166918851832332336318, 1.92069351415047393407405641465, 2.35408784622799217644785882796, 2.61337794086476379827969693424, 3.21883290576343141449284361679, 3.25300036473252915884705249000, 3.34581264028070520781275344117, 3.82795662696126431535837649972, 3.97392469223757199181451429030, 4.46981627935296430970816564571, 4.48461239067124809059727420586, 4.51708024808750445496230570398, 5.14021515441890065114126707089, 5.31289648926282314230319992294, 5.46139802300402566936098390033, 5.52857228721488793390946439266, 6.03131209356966241246113251243, 6.18063988834624981017115662548, 6.30777428209475975266412238365, 6.46243375406920911220503645520, 6.79384640444496381011296952785
