| L(s) = 1 | + 4-s − 2·7-s − 12·13-s + 4·16-s − 5·25-s − 2·28-s + 6·31-s − 4·37-s − 4·43-s − 11·49-s − 12·52-s + 24·61-s + 11·64-s + 20·67-s − 4·79-s + 24·91-s − 48·97-s − 5·100-s + 6·103-s − 28·109-s − 8·112-s − 17·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 3.32·13-s + 16-s − 25-s − 0.377·28-s + 1.07·31-s − 0.657·37-s − 0.609·43-s − 1.57·49-s − 1.66·52-s + 3.07·61-s + 11/8·64-s + 2.44·67-s − 0.450·79-s + 2.51·91-s − 4.87·97-s − 1/2·100-s + 0.591·103-s − 2.68·109-s − 0.755·112-s − 1.54·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3053494770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3053494770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 17 T^{2} + 168 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 41 T^{2} + 1152 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 38 T^{2} + 603 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 67 T^{2} + 2808 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 34 T^{2} - 1053 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 58 T^{2} - 117 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 106 T^{2} + 4347 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−7.75140953095226141983271967862, −7.66008300448257133868180718382, −7.07532262974005055142985997225, −7.02032549961440959918061110795, −6.80963969760706032240848351285, −6.68700996823244139425876799979, −6.44996414154175575816648554645, −6.19502444382945117239906528656, −5.58945848832521642738595927511, −5.52267600196777202611959129280, −5.31723219767917975453818055248, −5.09696565639077180913743624516, −4.95943732823055040046122508188, −4.40168858837561560161912127880, −4.28254387845320465658403683424, −3.88553356712731520371240458308, −3.50076372807318836771432365393, −3.46694001175117970725885928889, −2.85320460877019664862127618499, −2.53909191092273648893652756694, −2.34803643237193973464140030634, −2.34281753852418537369298031375, −1.51752136278362840118567896613, −1.17217951323115637267547343748, −0.16878747350830602566947762291,
0.16878747350830602566947762291, 1.17217951323115637267547343748, 1.51752136278362840118567896613, 2.34281753852418537369298031375, 2.34803643237193973464140030634, 2.53909191092273648893652756694, 2.85320460877019664862127618499, 3.46694001175117970725885928889, 3.50076372807318836771432365393, 3.88553356712731520371240458308, 4.28254387845320465658403683424, 4.40168858837561560161912127880, 4.95943732823055040046122508188, 5.09696565639077180913743624516, 5.31723219767917975453818055248, 5.52267600196777202611959129280, 5.58945848832521642738595927511, 6.19502444382945117239906528656, 6.44996414154175575816648554645, 6.68700996823244139425876799979, 6.80963969760706032240848351285, 7.02032549961440959918061110795, 7.07532262974005055142985997225, 7.66008300448257133868180718382, 7.75140953095226141983271967862
