L(s) = 1 | − 2·3-s − 4-s − 2·5-s + 2·8-s − 2·9-s + 6·11-s + 2·12-s + 4·15-s + 16-s − 8·19-s + 2·20-s + 4·23-s − 4·24-s − 6·25-s + 10·27-s + 6·29-s − 4·32-s − 12·33-s + 2·36-s − 6·37-s − 4·40-s + 12·43-s − 6·44-s + 4·45-s − 2·48-s + 2·49-s − 12·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.707·8-s − 2/3·9-s + 1.80·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s − 1.83·19-s + 0.447·20-s + 0.834·23-s − 0.816·24-s − 6/5·25-s + 1.92·27-s + 1.11·29-s − 0.707·32-s − 2.08·33-s + 1/3·36-s − 0.986·37-s − 0.632·40-s + 1.82·43-s − 0.904·44-s + 0.596·45-s − 0.288·48-s + 2/7·49-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2897979046\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2897979046\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−19.9621988029, −19.9292301410, −19.1494492613, −19.0177567201, −17.6761697408, −17.4469801976, −16.9473294095, −16.6317948443, −15.7312805996, −15.0693535709, −14.2197849424, −13.9795262067, −12.8595646329, −11.9583441737, −11.9362311444, −10.8489466872, −10.7173409989, −9.28607198826, −8.69568671187, −7.81910395524, −6.65330731261, −5.99911855172, −4.74199315541, −3.90229547123,
3.90229547123, 4.74199315541, 5.99911855172, 6.65330731261, 7.81910395524, 8.69568671187, 9.28607198826, 10.7173409989, 10.8489466872, 11.9362311444, 11.9583441737, 12.8595646329, 13.9795262067, 14.2197849424, 15.0693535709, 15.7312805996, 16.6317948443, 16.9473294095, 17.4469801976, 17.6761697408, 19.0177567201, 19.1494492613, 19.9292301410, 19.9621988029