L(s) = 1 | − 3-s + 4-s − 7-s − 12-s − 3·13-s + 21-s + 25-s + 27-s − 28-s + 2·37-s + 3·39-s + 49-s − 3·52-s − 64-s − 67-s + 2·73-s − 75-s − 3·79-s − 81-s + 84-s + 3·91-s + 100-s + 108-s − 3·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
L(s) = 1 | − 3-s + 4-s − 7-s − 12-s − 3·13-s + 21-s + 25-s + 27-s − 28-s + 2·37-s + 3·39-s + 49-s − 3·52-s − 64-s − 67-s + 2·73-s − 75-s − 3·79-s − 81-s + 84-s + 3·91-s + 100-s + 108-s − 3·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2786522884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2786522884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.36833704050682452832322795010, −13.55016410918758895210459647472, −12.72661832174701726040763085903, −12.61525216977882003906043806501, −11.91496355477986837445357335560, −11.79772524475894385983329721976, −11.02828773337518437619578361241, −10.63744028703307335460013897015, −9.896913020367610485625389816851, −9.678190572919910280677778371520, −8.971057301257502278056761614457, −7.997035025629676928244448689468, −7.19538498646848317933120534875, −7.05344265564969174407287336327, −6.35795379858912165737410810501, −5.76006380869294265534927873326, −5.00533503922753385933189223635, −4.41656836049261226463945647116, −2.87499744189534508364905084632, −2.51158045381864168173649537375,
2.51158045381864168173649537375, 2.87499744189534508364905084632, 4.41656836049261226463945647116, 5.00533503922753385933189223635, 5.76006380869294265534927873326, 6.35795379858912165737410810501, 7.05344265564969174407287336327, 7.19538498646848317933120534875, 7.997035025629676928244448689468, 8.971057301257502278056761614457, 9.678190572919910280677778371520, 9.896913020367610485625389816851, 10.63744028703307335460013897015, 11.02828773337518437619578361241, 11.79772524475894385983329721976, 11.91496355477986837445357335560, 12.61525216977882003906043806501, 12.72661832174701726040763085903, 13.55016410918758895210459647472, 14.36833704050682452832322795010