| L(s) = 1 | + 2·3-s + 2·9-s − 12·11-s − 8·23-s + 8·25-s + 6·27-s − 24·33-s + 16·37-s − 32·47-s + 16·49-s − 4·59-s − 16·61-s − 16·69-s − 8·71-s − 8·73-s + 16·75-s + 11·81-s − 20·83-s − 16·97-s − 24·99-s + 12·107-s + 16·109-s + 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2/3·9-s − 3.61·11-s − 1.66·23-s + 8/5·25-s + 1.15·27-s − 4.17·33-s + 2.63·37-s − 4.66·47-s + 16/7·49-s − 0.520·59-s − 2.04·61-s − 1.92·69-s − 0.949·71-s − 0.936·73-s + 1.84·75-s + 11/9·81-s − 2.19·83-s − 1.62·97-s − 2.41·99-s + 1.16·107-s + 1.53·109-s + 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.337383290270648066727179239920, −7.953166084353976866371216715996, −7.75214537122172409658378264757, −7.52962059160023067121229253521, −7.52827577989536865755828693224, −7.09850355644598100497572677221, −6.75303342761592234979106409086, −6.55119063926738476411848790961, −5.96930643351936550636399437278, −5.91212426319124269254500108422, −5.87214610005393243535764031529, −5.19077358503324674437993700634, −5.08143475494559960397632849984, −4.86746939517290063123510886470, −4.53068040854485427414322890199, −4.16387851453222514851865124053, −4.13616513667555005208236384816, −3.24550860816726745145212846961, −3.00485006546409638162381684884, −2.99148012092831753680283263274, −2.77065523905238320012856226399, −2.27482523376426099150410881987, −1.93774838897034715153343024495, −1.40555307368690100908531951299, −0.40534845052472367622557660495,
0.40534845052472367622557660495, 1.40555307368690100908531951299, 1.93774838897034715153343024495, 2.27482523376426099150410881987, 2.77065523905238320012856226399, 2.99148012092831753680283263274, 3.00485006546409638162381684884, 3.24550860816726745145212846961, 4.13616513667555005208236384816, 4.16387851453222514851865124053, 4.53068040854485427414322890199, 4.86746939517290063123510886470, 5.08143475494559960397632849984, 5.19077358503324674437993700634, 5.87214610005393243535764031529, 5.91212426319124269254500108422, 5.96930643351936550636399437278, 6.55119063926738476411848790961, 6.75303342761592234979106409086, 7.09850355644598100497572677221, 7.52827577989536865755828693224, 7.52962059160023067121229253521, 7.75214537122172409658378264757, 7.953166084353976866371216715996, 8.337383290270648066727179239920
