| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 2·16-s + 2·19-s − 20-s − 22-s − 2·24-s + 26-s − 27-s + 30-s − 2·32-s + 33-s + 2·37-s − 2·38-s − 39-s + 2·40-s + 44-s + 2·48-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 2·16-s + 2·19-s − 20-s − 22-s − 2·24-s + 26-s − 27-s + 30-s − 2·32-s + 33-s + 2·37-s − 2·38-s − 39-s + 2·40-s + 44-s + 2·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6197750068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6197750068\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.43110305780675066030312246768, −9.868351095495537992763089265994, −9.503887657488290941450810624542, −9.432149969288258726472015327291, −8.968412486621384362536473701498, −8.477507206788408641879294532296, −8.118696818140407307513335041412, −7.72873036252335400225208718436, −7.21425743079743222797291623599, −7.20878344629677365277336471919, −6.22175325819062784340835775617, −6.12109368317711078851245996097, −5.36499606257001995336925973340, −4.85841572712301605428615337350, −3.85018869333763373901578547749, −3.74856848021277261431932612987, −2.92812338595745336661144565351, −2.79417238625653623392222431050, −1.97736312379105568042916221952, −0.910952507530333710820793795314,
0.910952507530333710820793795314, 1.97736312379105568042916221952, 2.79417238625653623392222431050, 2.92812338595745336661144565351, 3.74856848021277261431932612987, 3.85018869333763373901578547749, 4.85841572712301605428615337350, 5.36499606257001995336925973340, 6.12109368317711078851245996097, 6.22175325819062784340835775617, 7.20878344629677365277336471919, 7.21425743079743222797291623599, 7.72873036252335400225208718436, 8.118696818140407307513335041412, 8.477507206788408641879294532296, 8.968412486621384362536473701498, 9.432149969288258726472015327291, 9.503887657488290941450810624542, 9.868351095495537992763089265994, 10.43110305780675066030312246768
