| L(s) = 1 | + 4-s − 4·7-s − 3·9-s + 4·13-s − 3·16-s − 4·19-s + 10·25-s − 4·28-s + 4·31-s − 3·36-s + 4·37-s − 12·43-s + 2·49-s + 4·52-s + 12·61-s + 12·63-s − 7·64-s − 12·73-s − 4·76-s + 8·79-s + 9·81-s − 16·91-s + 4·97-s + 10·100-s + 12·103-s − 12·109-s + 12·112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 9-s + 1.10·13-s − 3/4·16-s − 0.917·19-s + 2·25-s − 0.755·28-s + 0.718·31-s − 1/2·36-s + 0.657·37-s − 1.82·43-s + 2/7·49-s + 0.554·52-s + 1.53·61-s + 1.51·63-s − 7/8·64-s − 1.40·73-s − 0.458·76-s + 0.900·79-s + 81-s − 1.67·91-s + 0.406·97-s + 100-s + 1.18·103-s − 1.14·109-s + 1.13·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6364706559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6364706559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 241 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−13.13574808813755465520461448111, −12.79436034684729828270301416825, −11.92811151275482123934737928127, −11.37188049459337807838742715359, −10.79957829490858142714946697795, −10.20779122675769978234361342877, −9.392632585933952118237756926381, −8.703531204180469893411546526067, −8.304814903827842985922899528157, −6.99308969018060383581515623157, −6.48131976337285393293224274224, −6.05551703909288199729860189848, −4.83838587440655122228324789332, −3.52154536802576814629606243216, −2.69938163229631833131760768498,
2.69938163229631833131760768498, 3.52154536802576814629606243216, 4.83838587440655122228324789332, 6.05551703909288199729860189848, 6.48131976337285393293224274224, 6.99308969018060383581515623157, 8.304814903827842985922899528157, 8.703531204180469893411546526067, 9.392632585933952118237756926381, 10.20779122675769978234361342877, 10.79957829490858142714946697795, 11.37188049459337807838742715359, 11.92811151275482123934737928127, 12.79436034684729828270301416825, 13.13574808813755465520461448111
