| L(s) = 1 | + 4·5-s − 12·13-s − 4·17-s + 14·25-s − 16·29-s + 10·37-s + 16·41-s + 7·49-s + 20·53-s − 4·61-s − 48·65-s − 18·73-s − 16·85-s − 8·89-s − 24·97-s − 32·101-s + 30·109-s − 8·113-s + 22·121-s + 64·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 3.32·13-s − 0.970·17-s + 14/5·25-s − 2.97·29-s + 1.64·37-s + 2.49·41-s + 49-s + 2.74·53-s − 0.512·61-s − 5.95·65-s − 2.10·73-s − 1.73·85-s − 0.847·89-s − 2.43·97-s − 3.18·101-s + 2.87·109-s − 0.752·113-s + 2·121-s + 5.72·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4807419535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4807419535\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 17 T^{2} - 72 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 38 T^{2} + 915 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 41 T^{2} + 720 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 10 T^{2} - 2109 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 34 T^{2} - 2325 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 113 T^{2} + 8280 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 137 T^{2} + 12528 T^{4} - 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.60894862627344963086802856888, −6.06531566638181000292810112512, −5.98487230293476963883765056367, −5.87138455125239235207049391605, −5.65703710464808570382182365335, −5.58489695497745347135130207260, −5.32076220540788736967137314045, −4.98366375872647667003442306565, −4.86678870953405685603648338888, −4.45680248758202016078811165648, −4.45576113070818613452030450753, −4.38916092719926051960392024260, −3.83403591038429995640858068678, −3.75300965254489718132609129108, −3.40343289336921942885534594556, −2.89301023408079989086320186896, −2.69476443677215908255942533916, −2.54458142369602784404720457801, −2.35551527251111893178359595728, −2.21101294889662626362357440988, −2.05640372895073725800809543615, −1.38299253034354290765806159699, −1.30732487934328233790344544238, −0.78318447277551706593695661878, −0.11736500076414001011988671696,
0.11736500076414001011988671696, 0.78318447277551706593695661878, 1.30732487934328233790344544238, 1.38299253034354290765806159699, 2.05640372895073725800809543615, 2.21101294889662626362357440988, 2.35551527251111893178359595728, 2.54458142369602784404720457801, 2.69476443677215908255942533916, 2.89301023408079989086320186896, 3.40343289336921942885534594556, 3.75300965254489718132609129108, 3.83403591038429995640858068678, 4.38916092719926051960392024260, 4.45576113070818613452030450753, 4.45680248758202016078811165648, 4.86678870953405685603648338888, 4.98366375872647667003442306565, 5.32076220540788736967137314045, 5.58489695497745347135130207260, 5.65703710464808570382182365335, 5.87138455125239235207049391605, 5.98487230293476963883765056367, 6.06531566638181000292810112512, 6.60894862627344963086802856888
