L(s) = 1 | − 4·13-s + 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | − 4·13-s + 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9122044279\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9122044279\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.29596244471888632877575886245, −6.25443319809882618780658313562, −5.67988046107564714656975990578, −5.66979806296969947094178306581, −5.39124956076113367634690634621, −5.05493350352688407288564698729, −5.01615814203792437299528726007, −4.90449726426965376192210600265, −4.78952659669949514086963591717, −4.43822268827050100980762420462, −4.25824456212995958992485879422, −4.04770330587375452454865044459, −3.86226711840103339051260172007, −3.60535296727699148717555715829, −3.20289258474364028574680896810, −3.08710023255538321158036724516, −2.79469070822809598425280377879, −2.41360974141554376186838932514, −2.39260388633025422999209861023, −2.31431995466167475123812049611, −2.10734300108169582526941406054, −1.65719099457495338257020987190, −1.03496738874621367978271446123, −1.00610473011614275017122542910, −0.39850318103113549641196065899,
0.39850318103113549641196065899, 1.00610473011614275017122542910, 1.03496738874621367978271446123, 1.65719099457495338257020987190, 2.10734300108169582526941406054, 2.31431995466167475123812049611, 2.39260388633025422999209861023, 2.41360974141554376186838932514, 2.79469070822809598425280377879, 3.08710023255538321158036724516, 3.20289258474364028574680896810, 3.60535296727699148717555715829, 3.86226711840103339051260172007, 4.04770330587375452454865044459, 4.25824456212995958992485879422, 4.43822268827050100980762420462, 4.78952659669949514086963591717, 4.90449726426965376192210600265, 5.01615814203792437299528726007, 5.05493350352688407288564698729, 5.39124956076113367634690634621, 5.66979806296969947094178306581, 5.67988046107564714656975990578, 6.25443319809882618780658313562, 6.29596244471888632877575886245