| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 5-s + 4·6-s − 3·9-s − 2·10-s − 4·12-s + 8·13-s − 2·15-s − 4·16-s + 6·18-s + 2·20-s − 4·25-s − 16·26-s + 14·27-s + 4·30-s + 14·31-s + 8·32-s − 6·36-s + 6·37-s − 16·39-s − 16·41-s − 12·43-s − 3·45-s + 8·48-s − 10·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s + 1.63·6-s − 9-s − 0.632·10-s − 1.15·12-s + 2.21·13-s − 0.516·15-s − 16-s + 1.41·18-s + 0.447·20-s − 4/5·25-s − 3.13·26-s + 2.69·27-s + 0.730·30-s + 2.51·31-s + 1.41·32-s − 36-s + 0.986·37-s − 2.56·39-s − 2.49·41-s − 1.82·43-s − 0.447·45-s + 1.15·48-s − 1.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.615368948155826307952486308246, −8.603539619290756001226038948684, −8.168094638565977111025666477553, −7.67961078759111624265472503121, −6.71835585937068472809547273693, −6.36261389471308870138602900888, −6.16982430614728959932450070274, −5.70487824961827828764647644739, −4.82892009975420457193015711650, −4.60848682112829325490586806192, −3.37375563370605936780674564268, −2.99655198047806765916299649806, −1.80573857298709820434769765633, −1.12915794459905362466023426246, 0,
1.12915794459905362466023426246, 1.80573857298709820434769765633, 2.99655198047806765916299649806, 3.37375563370605936780674564268, 4.60848682112829325490586806192, 4.82892009975420457193015711650, 5.70487824961827828764647644739, 6.16982430614728959932450070274, 6.36261389471308870138602900888, 6.71835585937068472809547273693, 7.67961078759111624265472503121, 8.168094638565977111025666477553, 8.603539619290756001226038948684, 8.615368948155826307952486308246
