| L(s) = 1 | + 4-s − 2·5-s + 9-s − 2·20-s + 25-s − 6·29-s + 36-s − 2·41-s − 2·45-s − 2·49-s − 6·61-s − 64-s + 4·89-s + 100-s − 4·101-s − 4·109-s − 6·116-s + 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 4-s − 2·5-s + 9-s − 2·20-s + 25-s − 6·29-s + 36-s − 2·41-s − 2·45-s − 2·49-s − 6·61-s − 64-s + 4·89-s + 100-s − 4·101-s − 4·109-s − 6·116-s + 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3069300810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3069300810\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−7.25051909023950680412538339688, −7.06656770211473777228857460989, −6.70819433214857338657820417591, −6.61876952284989624106542717003, −6.28857878887941272030148867303, −6.12947245837479072340110608020, −5.72373977554336991486304502272, −5.71288777379053705001596430561, −5.32667596367070540388568609379, −5.17900827712441822337566401955, −4.82196709282594845077544206950, −4.53137574473272320847030183592, −4.39777872334338937201172873753, −4.07132290572644014598031320381, −3.87151233419364177439080981761, −3.65380538191506123414068692340, −3.56956795292172264680932431740, −3.08842417650883422307996263264, −2.92794546034471601862078544240, −2.83421344021656924349063129716, −1.85297912758672435225675914058, −1.76400738882299278238276240963, −1.74038011533462481330257198059, −1.66805206461756306521985861252, −0.33659931466699805046730468808,
0.33659931466699805046730468808, 1.66805206461756306521985861252, 1.74038011533462481330257198059, 1.76400738882299278238276240963, 1.85297912758672435225675914058, 2.83421344021656924349063129716, 2.92794546034471601862078544240, 3.08842417650883422307996263264, 3.56956795292172264680932431740, 3.65380538191506123414068692340, 3.87151233419364177439080981761, 4.07132290572644014598031320381, 4.39777872334338937201172873753, 4.53137574473272320847030183592, 4.82196709282594845077544206950, 5.17900827712441822337566401955, 5.32667596367070540388568609379, 5.71288777379053705001596430561, 5.72373977554336991486304502272, 6.12947245837479072340110608020, 6.28857878887941272030148867303, 6.61876952284989624106542717003, 6.70819433214857338657820417591, 7.06656770211473777228857460989, 7.25051909023950680412538339688
