| L(s) = 1 | − 4·7-s − 2·9-s + 4·13-s + 8·19-s − 6·23-s + 25-s + 12·29-s + 12·41-s + 20·43-s − 2·49-s + 8·63-s − 4·67-s + 4·73-s − 16·79-s − 5·81-s − 12·83-s − 16·91-s + 12·101-s − 28·103-s + 12·107-s − 8·117-s − 22·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 2/3·9-s + 1.10·13-s + 1.83·19-s − 1.25·23-s + 1/5·25-s + 2.22·29-s + 1.87·41-s + 3.04·43-s − 2/7·49-s + 1.00·63-s − 0.488·67-s + 0.468·73-s − 1.80·79-s − 5/9·81-s − 1.31·83-s − 1.67·91-s + 1.19·101-s − 2.75·103-s + 1.16·107-s − 0.739·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617600825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617600825\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.226596894071287240955393392679, −7.71273110823499542846308181544, −7.41584760680109602943805602282, −6.79622880231902033736808183873, −6.27087624192875571051851265258, −6.07443665403472328256818219598, −5.67364018466611088817999707244, −5.17758557239314882693293351502, −4.22981234422099023252949896613, −4.12250433368686324236236368171, −3.33310298408427540540082175752, −2.76929890617261215013507568311, −2.71325890358691787487172487829, −1.36646892315183138105046537265, −0.65658545895941377481149023623,
0.65658545895941377481149023623, 1.36646892315183138105046537265, 2.71325890358691787487172487829, 2.76929890617261215013507568311, 3.33310298408427540540082175752, 4.12250433368686324236236368171, 4.22981234422099023252949896613, 5.17758557239314882693293351502, 5.67364018466611088817999707244, 6.07443665403472328256818219598, 6.27087624192875571051851265258, 6.79622880231902033736808183873, 7.41584760680109602943805602282, 7.71273110823499542846308181544, 8.226596894071287240955393392679
