| L(s) = 1 | + 180·5-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 + 1.06e7·49-s − 2.12e6·53-s + 3.06e7·61-s − 5.88e6·65-s + 3.78e7·73-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 + 4.01e8·121-s − 2.08e8·125-s + 127-s + 131-s + 137-s + 139-s + 4.57e8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.287·5-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 + 1.85·49-s − 0.269·53-s + 2.21·61-s − 0.329·65-s + 1.33·73-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 + 1.87·121-s − 0.852·125-s + 1.03·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.002950075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.002950075\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 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
−11.91969413309600510255756934173, −11.52049924013357489567843432359, −10.48361584951210614114453546207, −10.30210070366609242039848691761, −9.904438474711317518988536627183, −9.366593568856988840199752609049, −8.564156657541850557346665924347, −8.291952700845458069695877935627, −7.51413129262589969668738563204, −7.14547721249726636413502505342, −6.41359828006030888892688038396, −5.91101725817617458703989798770, −5.12194301776496160892419240771, −4.84758703406020830103838217920, −3.91784902888304907540352047693, −3.34042762610759719727056075875, −2.46764524575063832550864509418, −2.07103335954709203375990945327, −1.04104138746452616640914780839, −0.50341498390618007172972915474,
0.50341498390618007172972915474, 1.04104138746452616640914780839, 2.07103335954709203375990945327, 2.46764524575063832550864509418, 3.34042762610759719727056075875, 3.91784902888304907540352047693, 4.84758703406020830103838217920, 5.12194301776496160892419240771, 5.91101725817617458703989798770, 6.41359828006030888892688038396, 7.14547721249726636413502505342, 7.51413129262589969668738563204, 8.291952700845458069695877935627, 8.564156657541850557346665924347, 9.366593568856988840199752609049, 9.904438474711317518988536627183, 10.30210070366609242039848691761, 10.48361584951210614114453546207, 11.52049924013357489567843432359, 11.91969413309600510255756934173
