| L(s) = 1 | + 4-s − 6·13-s + 25-s + 2·49-s − 6·52-s − 2·61-s − 64-s + 100-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
| L(s) = 1 | + 4-s − 6·13-s + 25-s + 2·49-s − 6·52-s − 2·61-s − 64-s + 100-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4495315498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4495315498\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.12821107160183185585893551568, −6.71960862007963364035735181838, −6.57532934662455015950951018342, −6.49725292776363356088912522476, −5.97924060921549831247525692325, −5.71290930797732832286846249316, −5.64464347314300908626297058920, −5.37077426520007940626515036357, −5.09482030385721969592910086986, −4.92498080651624038295448019293, −4.65344081520144472969315772992, −4.62597409388878188053652233171, −4.48801351537193899049276887311, −4.05065653436278199447280501935, −3.83415364223933840631437076019, −3.29767632926102987660041848069, −3.18975803668041914658563951657, −2.72277560087324283984708393983, −2.64503752200081219583980833089, −2.43109394178940768097480918369, −2.27115224643207488907508161310, −2.19635934830532508241228443941, −1.47878946688837208851053819259, −1.33298622868522307760368747257, −0.34606435829644916010261749344,
0.34606435829644916010261749344, 1.33298622868522307760368747257, 1.47878946688837208851053819259, 2.19635934830532508241228443941, 2.27115224643207488907508161310, 2.43109394178940768097480918369, 2.64503752200081219583980833089, 2.72277560087324283984708393983, 3.18975803668041914658563951657, 3.29767632926102987660041848069, 3.83415364223933840631437076019, 4.05065653436278199447280501935, 4.48801351537193899049276887311, 4.62597409388878188053652233171, 4.65344081520144472969315772992, 4.92498080651624038295448019293, 5.09482030385721969592910086986, 5.37077426520007940626515036357, 5.64464347314300908626297058920, 5.71290930797732832286846249316, 5.97924060921549831247525692325, 6.49725292776363356088912522476, 6.57532934662455015950951018342, 6.71960862007963364035735181838, 7.12821107160183185585893551568
