| L(s) = 1 | + 3-s − 4·4-s − 2·9-s − 11-s − 4·12-s + 12·16-s − 12·17-s − 25-s − 5·27-s − 12·29-s + 10·31-s − 33-s + 8·36-s + 22·37-s + 12·41-s + 4·44-s + 12·48-s + 49-s − 12·51-s − 32·64-s + 10·67-s + 48·68-s − 75-s + 81-s + 24·83-s − 12·87-s + 10·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s − 2/3·9-s − 0.301·11-s − 1.15·12-s + 3·16-s − 2.91·17-s − 1/5·25-s − 0.962·27-s − 2.22·29-s + 1.79·31-s − 0.174·33-s + 4/3·36-s + 3.61·37-s + 1.87·41-s + 0.603·44-s + 1.73·48-s + 1/7·49-s − 1.68·51-s − 4·64-s + 1.22·67-s + 5.82·68-s − 0.115·75-s + 1/9·81-s + 2.63·83-s − 1.28·87-s + 1.03·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−8.198299351340584631525150242215, −7.79780189933062788077562645237, −7.79543673470654027074280114997, −6.81062854234304516450450908396, −6.19954355560081421436202244959, −5.89237731397243587720148634695, −5.35059430166220814299615131856, −4.69940224529243356978622440045, −4.31297732989480839195031476377, −4.09981281561803996499645678286, −3.45131480487179814577542455626, −2.52489627324288051568453078479, −2.33532150394146050566081783680, −0.896810126936371404627589617928, 0,
0.896810126936371404627589617928, 2.33532150394146050566081783680, 2.52489627324288051568453078479, 3.45131480487179814577542455626, 4.09981281561803996499645678286, 4.31297732989480839195031476377, 4.69940224529243356978622440045, 5.35059430166220814299615131856, 5.89237731397243587720148634695, 6.19954355560081421436202244959, 6.81062854234304516450450908396, 7.79543673470654027074280114997, 7.79780189933062788077562645237, 8.198299351340584631525150242215
