| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s + 9-s − 3·10-s − 12-s − 3·15-s + 16-s − 18-s + 6·19-s + 3·20-s + 12·23-s + 24-s + 8·25-s − 27-s + 29-s + 3·30-s − 32-s + 36-s − 6·38-s − 3·40-s − 6·43-s + 3·45-s − 12·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s − 0.774·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 2.50·23-s + 0.204·24-s + 8/5·25-s − 0.192·27-s + 0.185·29-s + 0.547·30-s − 0.176·32-s + 1/6·36-s − 0.973·38-s − 0.474·40-s − 0.914·43-s + 0.447·45-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699651152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699651152\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 152 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.478132576237499548746706119317, −8.207685693655239756173154138964, −7.59779669769229146224591246919, −6.85124852212508370787937953976, −6.75453237485395520047942270054, −6.51962891242773868964115648612, −5.52451827726392759269121259600, −5.35608386022049082089079586596, −5.07961053891977601175214247532, −4.34061539587989508610527705279, −3.26456392803145582594835626349, −3.06006765684046179868468076581, −2.22182573033452620791687510870, −1.44134350435519549997788741184, −0.889865252469933683858923373717,
0.889865252469933683858923373717, 1.44134350435519549997788741184, 2.22182573033452620791687510870, 3.06006765684046179868468076581, 3.26456392803145582594835626349, 4.34061539587989508610527705279, 5.07961053891977601175214247532, 5.35608386022049082089079586596, 5.52451827726392759269121259600, 6.51962891242773868964115648612, 6.75453237485395520047942270054, 6.85124852212508370787937953976, 7.59779669769229146224591246919, 8.207685693655239756173154138964, 8.478132576237499548746706119317
