| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3·11-s + 12-s + 16-s + 4·17-s − 18-s + 19-s − 3·22-s − 24-s − 25-s + 27-s − 32-s + 3·33-s − 4·34-s + 36-s − 38-s − 11·43-s + 3·44-s + 48-s + 49-s + 50-s + 4·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.639·22-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.176·32-s + 0.522·33-s − 0.685·34-s + 1/6·36-s − 0.162·38-s − 1.67·43-s + 0.452·44-s + 0.144·48-s + 1/7·49-s + 0.141·50-s + 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2878848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2878848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−7.45578989288019135896077756618, −7.01662379241515117834417453348, −6.69677239834277945706983087327, −6.16780392122312587213348990985, −5.87370041008452867744729062589, −5.19018087010199286506133400362, −4.91856003682954245120569814580, −4.05728529269364458327803250519, −3.94258878630875933137679805516, −3.11312994413698073780781142937, −3.00950790433389335019302713929, −2.20515413095174031848826156180, −1.47551861632724679570109027241, −1.23392273324238793066202434858, 0,
1.23392273324238793066202434858, 1.47551861632724679570109027241, 2.20515413095174031848826156180, 3.00950790433389335019302713929, 3.11312994413698073780781142937, 3.94258878630875933137679805516, 4.05728529269364458327803250519, 4.91856003682954245120569814580, 5.19018087010199286506133400362, 5.87370041008452867744729062589, 6.16780392122312587213348990985, 6.69677239834277945706983087327, 7.01662379241515117834417453348, 7.45578989288019135896077756618
