L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·9-s + 4·13-s + 5·16-s − 6·17-s + 4·18-s − 8·19-s + 2·25-s − 8·26-s − 6·32-s + 12·34-s − 6·36-s + 16·38-s − 8·43-s + 14·49-s − 4·50-s + 12·52-s + 12·53-s + 24·59-s + 7·64-s − 8·67-s − 18·68-s + 8·72-s − 24·76-s − 5·81-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s + 1.10·13-s + 5/4·16-s − 1.45·17-s + 0.942·18-s − 1.83·19-s + 2/5·25-s − 1.56·26-s − 1.06·32-s + 2.05·34-s − 36-s + 2.59·38-s − 1.21·43-s + 2·49-s − 0.565·50-s + 1.66·52-s + 1.64·53-s + 3.12·59-s + 7/8·64-s − 0.977·67-s − 2.18·68-s + 0.942·72-s − 2.75·76-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3093859603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3093859603\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−17.06167896434940375430658952132, −16.61092041346761771305940008744, −15.99776201174963910559487020643, −15.27285023183387587548225226051, −15.00864147381443500873456504151, −14.12719507516391158951572154901, −13.23246082696141745797793564410, −12.84960809136784896974761597774, −11.62104328433788920008879166606, −11.48138420354830170197930020217, −10.44767684414425161642458569644, −10.42251693501845675043344465453, −9.112705956177541863647570903426, −8.517394393692955508869058389691, −8.484102564279125782220274863224, −7.10604028804496855298145497397, −6.53747231897222211021137211021, −5.68070638546849464095536524868, −4.05533330791451947168279880511, −2.37714604467739077916333919542,
2.37714604467739077916333919542, 4.05533330791451947168279880511, 5.68070638546849464095536524868, 6.53747231897222211021137211021, 7.10604028804496855298145497397, 8.484102564279125782220274863224, 8.517394393692955508869058389691, 9.112705956177541863647570903426, 10.42251693501845675043344465453, 10.44767684414425161642458569644, 11.48138420354830170197930020217, 11.62104328433788920008879166606, 12.84960809136784896974761597774, 13.23246082696141745797793564410, 14.12719507516391158951572154901, 15.00864147381443500873456504151, 15.27285023183387587548225226051, 15.99776201174963910559487020643, 16.61092041346761771305940008744, 17.06167896434940375430658952132