L(s) = 1 | − 9-s + 8·13-s − 8·17-s + 25-s − 8·29-s + 8·37-s − 4·41-s − 2·49-s + 16·53-s + 4·61-s + 4·73-s + 81-s + 12·89-s − 20·97-s − 8·101-s + 4·109-s + 16·113-s − 8·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 2.21·13-s − 1.94·17-s + 1/5·25-s − 1.48·29-s + 1.31·37-s − 0.624·41-s − 2/7·49-s + 2.19·53-s + 0.512·61-s + 0.468·73-s + 1/9·81-s + 1.27·89-s − 2.03·97-s − 0.796·101-s + 0.383·109-s + 1.50·113-s − 0.739·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.849115622\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.849115622\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.404500107824392322127160166903, −7.77441264419006401140423487244, −7.18926638914786237994833871145, −6.85339619928887360057874840860, −6.28404737886842815600942616629, −6.04829656208433597478128855321, −5.57099577868031336468001305935, −5.01582318939639180218037564790, −4.37233895265217885024449377948, −3.89747012731281061247751281966, −3.63051576685478776731945960312, −2.83474565745167995209650669612, −2.21263756864543494116318571706, −1.59269691415396240921872407864, −0.65019890207007053831444091905,
0.65019890207007053831444091905, 1.59269691415396240921872407864, 2.21263756864543494116318571706, 2.83474565745167995209650669612, 3.63051576685478776731945960312, 3.89747012731281061247751281966, 4.37233895265217885024449377948, 5.01582318939639180218037564790, 5.57099577868031336468001305935, 6.04829656208433597478128855321, 6.28404737886842815600942616629, 6.85339619928887360057874840860, 7.18926638914786237994833871145, 7.77441264419006401140423487244, 8.404500107824392322127160166903