| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 3·9-s + 2·11-s + 2·12-s − 16-s − 2·17-s − 3·18-s + 6·19-s − 2·22-s − 6·24-s − 25-s − 4·27-s − 5·32-s − 4·33-s + 2·34-s − 3·36-s − 6·38-s − 14·41-s − 8·43-s − 2·44-s + 2·48-s + 2·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 9-s + 0.603·11-s + 0.577·12-s − 1/4·16-s − 0.485·17-s − 0.707·18-s + 1.37·19-s − 0.426·22-s − 1.22·24-s − 1/5·25-s − 0.769·27-s − 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/2·36-s − 0.973·38-s − 2.18·41-s − 1.21·43-s − 0.301·44-s + 0.288·48-s + 2/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.57058203818253197281768482198, −7.14208530046753208660020336733, −6.83530621471581364316344386979, −6.41005176080163568306446978225, −5.89443824717980479468779817353, −5.33659505692247112288538356386, −5.06051045154217671059803900195, −4.66242347146520972871041054491, −4.13072360398225293655004443930, −3.55729432869759313679055657725, −3.15511793561358484234928143704, −2.03483153664458381321364283439, −1.52790826194540348581362128578, −0.860409627779462174358550875201, 0,
0.860409627779462174358550875201, 1.52790826194540348581362128578, 2.03483153664458381321364283439, 3.15511793561358484234928143704, 3.55729432869759313679055657725, 4.13072360398225293655004443930, 4.66242347146520972871041054491, 5.06051045154217671059803900195, 5.33659505692247112288538356386, 5.89443824717980479468779817353, 6.41005176080163568306446978225, 6.83530621471581364316344386979, 7.14208530046753208660020336733, 7.57058203818253197281768482198
