L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 8-s − 10-s − 3·11-s − 10·13-s − 15-s − 16-s + 5·19-s − 3·22-s + 9·23-s − 24-s − 10·26-s − 27-s − 30-s − 10·31-s − 3·33-s + 37-s + 5·38-s − 10·39-s + 40-s − 18·41-s + 16·43-s + 9·46-s + 3·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 2.77·13-s − 0.258·15-s − 1/4·16-s + 1.14·19-s − 0.639·22-s + 1.87·23-s − 0.204·24-s − 1.96·26-s − 0.192·27-s − 0.182·30-s − 1.79·31-s − 0.522·33-s + 0.164·37-s + 0.811·38-s − 1.60·39-s + 0.158·40-s − 2.81·41-s + 2.43·43-s + 1.32·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.493675864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493675864\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.583349445371523017946238017070, −9.338952788673778718278910018398, −8.900167007576851669182415281404, −8.639786271306232522052338089798, −7.86690420206183059297713685927, −7.62201900451264564953591475851, −7.40222331275373960640187663321, −6.83306942599835031352132826012, −6.81029359105270310955149368595, −5.63271476678346637420463689687, −5.42140972191239109701616912053, −5.08915798257321739408074917488, −4.88629136510676719090207660902, −4.04767663942300464093416786516, −3.87074536342218035859977059138, −2.91854545247552529900806380580, −2.87091377713771768076655416324, −2.43240904203746776925018671566, −1.57234960949782816263250875200, −0.40367503742902975301103920312,
0.40367503742902975301103920312, 1.57234960949782816263250875200, 2.43240904203746776925018671566, 2.87091377713771768076655416324, 2.91854545247552529900806380580, 3.87074536342218035859977059138, 4.04767663942300464093416786516, 4.88629136510676719090207660902, 5.08915798257321739408074917488, 5.42140972191239109701616912053, 5.63271476678346637420463689687, 6.81029359105270310955149368595, 6.83306942599835031352132826012, 7.40222331275373960640187663321, 7.62201900451264564953591475851, 7.86690420206183059297713685927, 8.639786271306232522052338089798, 8.900167007576851669182415281404, 9.338952788673778718278910018398, 9.583349445371523017946238017070