| L(s) = 1 | − 2·4-s + 4·9-s − 8·11-s + 3·16-s − 4·19-s + 8·29-s − 4·31-s − 8·36-s − 8·41-s + 16·44-s + 4·49-s + 20·59-s + 8·61-s − 4·64-s − 16·71-s + 8·76-s − 28·79-s + 6·81-s + 8·89-s − 32·99-s − 24·101-s − 8·109-s − 16·116-s + 20·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s + 4/3·9-s − 2.41·11-s + 3/4·16-s − 0.917·19-s + 1.48·29-s − 0.718·31-s − 4/3·36-s − 1.24·41-s + 2.41·44-s + 4/7·49-s + 2.60·59-s + 1.02·61-s − 1/2·64-s − 1.89·71-s + 0.917·76-s − 3.15·79-s + 2/3·81-s + 0.847·89-s − 3.21·99-s − 2.38·101-s − 0.766·109-s − 1.48·116-s + 1.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2890866422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2890866422\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 710 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2634 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 204 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26186 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.65029793566055903355546308856, −6.35424620082476919540345432579, −6.18629492953428028726422166611, −5.78483252997752371096327570559, −5.71592599953374055210745278461, −5.38224302748975609646192112287, −5.18063091829708337009660101916, −5.10136191615120178248826050008, −5.07231044088994821649847385769, −4.52191808866164364804430068844, −4.30080619409569732522080765437, −4.28932651868748553973147135274, −4.07940059263292907150033875044, −3.80047590515608099156433508570, −3.53839405809838761439950205447, −2.99835697077606186585073568699, −2.93516964188152706749074709112, −2.86820236731229023909658896955, −2.35536341556365986370658436055, −2.14137728201475462284399007313, −1.88321315829624607037759863753, −1.41900352738111328028068348418, −1.15404701267860452455985041143, −0.68751852305849781152909956136, −0.12368224523285514056220162979,
0.12368224523285514056220162979, 0.68751852305849781152909956136, 1.15404701267860452455985041143, 1.41900352738111328028068348418, 1.88321315829624607037759863753, 2.14137728201475462284399007313, 2.35536341556365986370658436055, 2.86820236731229023909658896955, 2.93516964188152706749074709112, 2.99835697077606186585073568699, 3.53839405809838761439950205447, 3.80047590515608099156433508570, 4.07940059263292907150033875044, 4.28932651868748553973147135274, 4.30080619409569732522080765437, 4.52191808866164364804430068844, 5.07231044088994821649847385769, 5.10136191615120178248826050008, 5.18063091829708337009660101916, 5.38224302748975609646192112287, 5.71592599953374055210745278461, 5.78483252997752371096327570559, 6.18629492953428028726422166611, 6.35424620082476919540345432579, 6.65029793566055903355546308856
