| L(s) = 1 | + 4-s + 3·13-s + 2·25-s − 3·31-s − 37-s − 43-s + 3·52-s + 3·61-s − 64-s + 67-s + 79-s + 3·97-s + 2·100-s − 109-s − 2·121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + ⋯ |
| L(s) = 1 | + 4-s + 3·13-s + 2·25-s − 3·31-s − 37-s − 43-s + 3·52-s + 3·61-s − 64-s + 67-s + 79-s + 3·97-s + 2·100-s − 109-s − 2·121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.359276947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.359276947\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.873187281100141177641773149947, −8.488633838646607235813003965186, −8.093556139875731936879771478104, −7.80075252924838030355791552784, −7.15261515896159011067601935442, −6.84981517408924346700867204470, −6.77366855644605466750994712019, −6.29974580381200253090656770581, −5.93867407403977119652919655640, −5.56371046908188918456572519489, −5.01392596827841682014063889009, −4.96477954992769936883061203332, −3.86665799230948920283244248292, −3.79591428457002125219018750350, −3.55601717595189937183127828890, −3.00137710005122716160463668300, −2.40538823173229163531696321226, −1.88536883880065982094522310803, −1.46707539226908886210842240472, −0.929211114693954615932929221806,
0.929211114693954615932929221806, 1.46707539226908886210842240472, 1.88536883880065982094522310803, 2.40538823173229163531696321226, 3.00137710005122716160463668300, 3.55601717595189937183127828890, 3.79591428457002125219018750350, 3.86665799230948920283244248292, 4.96477954992769936883061203332, 5.01392596827841682014063889009, 5.56371046908188918456572519489, 5.93867407403977119652919655640, 6.29974580381200253090656770581, 6.77366855644605466750994712019, 6.84981517408924346700867204470, 7.15261515896159011067601935442, 7.80075252924838030355791552784, 8.093556139875731936879771478104, 8.488633838646607235813003965186, 8.873187281100141177641773149947
