| L(s) = 1 | + 180·5-s − 2.18e3·9-s + 3.27e4·13-s − 8.53e4·17-s − 7.56e5·25-s + 2.54e6·29-s + 4.52e6·37-s − 1.74e6·41-s − 3.93e5·45-s + 1.06e7·49-s − 2.12e6·53-s − 3.06e7·61-s + 5.88e6·65-s + 3.78e7·73-s + 4.78e6·81-s − 1.53e7·85-s − 1.79e8·89-s − 1.51e8·97-s + 3.37e8·101-s + 2.75e7·109-s − 1.85e8·113-s − 7.15e7·117-s + 4.01e8·121-s − 2.08e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.287·5-s − 1/3·9-s + 1.14·13-s − 1.02·17-s − 1.93·25-s + 3.59·29-s + 2.41·37-s − 0.617·41-s − 0.0959·45-s + 1.85·49-s − 0.269·53-s − 2.21·61-s + 0.329·65-s + 1.33·73-s + 1/9·81-s − 0.294·85-s − 2.86·89-s − 1.71·97-s + 3.24·101-s + 0.195·109-s − 1.13·113-s − 0.381·117-s + 1.87·121-s − 0.852·125-s + 1.03·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.282083601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282083601\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 18 p T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 217970 p^{2} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 401933330 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16358 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 42678 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20869012754 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 93790335362 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1270530 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1442671161170 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2262142 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 872694 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20613881044754 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 46987353734210 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1061694 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 252602261223790 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 15301010 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 727858867788434 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 770253450231170 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 18916354 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 100988299996754 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 252377493036434 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 89813214 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 75778238 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17663564458558892906729665674, −10.92331027106690402170520008737, −10.31493519699668690291686287187, −9.807115753948319195432182315175, −9.401717918600168033619578215770, −8.623849205015333757415804978052, −8.435221770366683672513051494272, −7.909163405288233887328323742015, −7.20810376766686490097042406009, −6.51101887454120145175148926990, −6.08503495102460841810916594249, −5.84501023397055437824002751095, −4.84522668961560352330140313436, −4.37989315803777759435320920620, −3.87562757559830239112091574006, −2.94189272429569635697947637394, −2.56096444042116705894423438857, −1.77223625393023371823959703720, −1.02725113268760986129137798978, −0.48186481734569619373559880614,
0.48186481734569619373559880614, 1.02725113268760986129137798978, 1.77223625393023371823959703720, 2.56096444042116705894423438857, 2.94189272429569635697947637394, 3.87562757559830239112091574006, 4.37989315803777759435320920620, 4.84522668961560352330140313436, 5.84501023397055437824002751095, 6.08503495102460841810916594249, 6.51101887454120145175148926990, 7.20810376766686490097042406009, 7.909163405288233887328323742015, 8.435221770366683672513051494272, 8.623849205015333757415804978052, 9.401717918600168033619578215770, 9.807115753948319195432182315175, 10.31493519699668690291686287187, 10.92331027106690402170520008737, 11.17663564458558892906729665674
