| L(s) = 1 | − 4-s − 2·7-s + 11-s + 3·13-s + 16-s + 17-s − 5·19-s − 7·23-s + 25-s + 2·28-s − 9·37-s + 19·41-s − 44-s − 3·49-s − 3·52-s − 2·53-s − 61-s − 64-s − 4·67-s − 68-s − 11·71-s + 8·73-s + 5·76-s − 2·77-s − 9·81-s − 12·83-s − 6·91-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 0.301·11-s + 0.832·13-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 1.45·23-s + 1/5·25-s + 0.377·28-s − 1.47·37-s + 2.96·41-s − 0.150·44-s − 3/7·49-s − 0.416·52-s − 0.274·53-s − 0.128·61-s − 1/8·64-s − 0.488·67-s − 0.121·68-s − 1.30·71-s + 0.936·73-s + 0.573·76-s − 0.227·77-s − 81-s − 1.31·83-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
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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
−8.696642079624937501854375917660, −8.379237936529884523653145670823, −7.73640633952606073847827682378, −7.35315014139389657264579361285, −6.67655596129607236416357430723, −6.11642768139625318831023972647, −6.00881595940255549707317830487, −5.34712453072282068214034289957, −4.53634730075534051254347497520, −4.11395115535224682280482922425, −3.68703812498954776306897258893, −2.99808135125263367248845353049, −2.22144675105971451204804045486, −1.29933314100272110073306953845, 0,
1.29933314100272110073306953845, 2.22144675105971451204804045486, 2.99808135125263367248845353049, 3.68703812498954776306897258893, 4.11395115535224682280482922425, 4.53634730075534051254347497520, 5.34712453072282068214034289957, 6.00881595940255549707317830487, 6.11642768139625318831023972647, 6.67655596129607236416357430723, 7.35315014139389657264579361285, 7.73640633952606073847827682378, 8.379237936529884523653145670823, 8.696642079624937501854375917660
