| L(s) = 1 | + 4·3-s − 12·7-s + 8·9-s − 48·21-s + 12·27-s − 8·31-s + 8·37-s + 8·43-s + 72·49-s − 24·61-s − 96·63-s − 16·67-s − 12·73-s + 23·81-s − 32·93-s + 52·97-s + 44·103-s + 32·111-s + 40·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s + 288·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4.53·7-s + 8/3·9-s − 10.4·21-s + 2.30·27-s − 1.43·31-s + 1.31·37-s + 1.21·43-s + 72/7·49-s − 3.07·61-s − 12.0·63-s − 1.95·67-s − 1.40·73-s + 23/9·81-s − 3.31·93-s + 5.27·97-s + 4.33·103-s + 3.03·111-s + 3.63·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 23.7·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700023238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700023238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 3518 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.60689976931494954792452632750, −7.39594522252576426712038542949, −7.34290803399957647415589062383, −7.21819065090895601978676571231, −6.75359766451144826774473286639, −6.31798232045585439324785017930, −6.26721286446570257336003727360, −6.18200349647624083788691126250, −6.04126992521340055942578711387, −5.70209234447232390769790061784, −5.37721151937370020053293826708, −4.69358348952496015804781533569, −4.52781797525554858933349514616, −4.44086421624489791051725399523, −3.91712875277850756595439968011, −3.50786974838379001124111800502, −3.36461889612625722427737669008, −3.35316540524734806991000923060, −3.07992668812418936133010503143, −2.88628410245204732300949769799, −2.51149917893159982814059512944, −2.06320607704949247415784799678, −1.94767599230010331403798451430, −0.964903326611418223531473243068, −0.36374035517925994833875715008,
0.36374035517925994833875715008, 0.964903326611418223531473243068, 1.94767599230010331403798451430, 2.06320607704949247415784799678, 2.51149917893159982814059512944, 2.88628410245204732300949769799, 3.07992668812418936133010503143, 3.35316540524734806991000923060, 3.36461889612625722427737669008, 3.50786974838379001124111800502, 3.91712875277850756595439968011, 4.44086421624489791051725399523, 4.52781797525554858933349514616, 4.69358348952496015804781533569, 5.37721151937370020053293826708, 5.70209234447232390769790061784, 6.04126992521340055942578711387, 6.18200349647624083788691126250, 6.26721286446570257336003727360, 6.31798232045585439324785017930, 6.75359766451144826774473286639, 7.21819065090895601978676571231, 7.34290803399957647415589062383, 7.39594522252576426712038542949, 7.60689976931494954792452632750
