| L(s) = 1 | − 17·16-s − 8·31-s − 472·61-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 1.06·16-s − 0.258·31-s − 7.73·61-s − 4·121-s + 0.00787·127-s + 0.00763·131-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 + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + 0.00414·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6843660893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6843660893\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 17 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 21118 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 550078 T^{4} + p^{8} T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 8065922 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 12619678 T^{4} + p^{8} T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 118 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2878 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 3847202 T^{4} + p^{8} T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
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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
−8.905540783836899547859258364014, −8.423910199680404325616157462590, −8.079842980441355550404663065878, −7.85132226455557793632681036504, −7.78083339993714620212889445127, −7.34932448818803496226290868625, −7.13162501031343482099412714156, −6.83470359526966996698559614028, −6.53662066308075619474913943166, −6.18726223284535473432464099101, −5.97385339340510507933544251691, −5.86308772050965908078271800714, −5.22016296116576028515421945105, −5.10460120676768696303338122180, −4.61530659453028256684999096811, −4.49388909844363532316251117371, −4.15707606957400268870748456939, −3.81373201045007655272614123957, −3.19714912121795200649681853331, −2.94385203799207676416137919646, −2.80711046425573059783359235721, −1.99420629929663501466532435632, −1.74148456731391894367844642438, −1.22303572408527344757818884820, −0.22981660182098228922443903012,
0.22981660182098228922443903012, 1.22303572408527344757818884820, 1.74148456731391894367844642438, 1.99420629929663501466532435632, 2.80711046425573059783359235721, 2.94385203799207676416137919646, 3.19714912121795200649681853331, 3.81373201045007655272614123957, 4.15707606957400268870748456939, 4.49388909844363532316251117371, 4.61530659453028256684999096811, 5.10460120676768696303338122180, 5.22016296116576028515421945105, 5.86308772050965908078271800714, 5.97385339340510507933544251691, 6.18726223284535473432464099101, 6.53662066308075619474913943166, 6.83470359526966996698559614028, 7.13162501031343482099412714156, 7.34932448818803496226290868625, 7.78083339993714620212889445127, 7.85132226455557793632681036504, 8.079842980441355550404663065878, 8.423910199680404325616157462590, 8.905540783836899547859258364014
