| L(s) = 1 | − 2-s + 3·3-s − 5-s − 3·6-s + 8-s + 3·9-s + 10-s + 2·11-s − 3·15-s − 16-s − 4·17-s − 3·18-s − 6·19-s − 2·22-s − 3·23-s + 3·24-s + 18·29-s + 3·30-s − 4·31-s + 6·33-s + 4·34-s + 4·37-s + 6·38-s − 40-s + 14·41-s − 10·43-s − 3·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 0.447·5-s − 1.22·6-s + 0.353·8-s + 9-s + 0.316·10-s + 0.603·11-s − 0.774·15-s − 1/4·16-s − 0.970·17-s − 0.707·18-s − 1.37·19-s − 0.426·22-s − 0.625·23-s + 0.612·24-s + 3.34·29-s + 0.547·30-s − 0.718·31-s + 1.04·33-s + 0.685·34-s + 0.657·37-s + 0.973·38-s − 0.158·40-s + 2.18·41-s − 1.52·43-s − 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773407774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773407774\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.97665083703086790483412661559, −10.67301765985316947394187938296, −10.16781257917926660068681781130, −9.624615551705869284953254924198, −9.335723948153828938297310453389, −8.756988663527870880694039205949, −8.493255312495664521477529628378, −8.211240608656270383576244000224, −8.019809309138138701606525226178, −6.99391885917245285038671016395, −6.98978401085185422934519850516, −6.26317338191743178486844825574, −5.70808504593567739297744596633, −4.53046448399197474587524858813, −4.47020055782771344255608591463, −3.82586841758677743731874991009, −3.13625436568813031171355036015, −2.45281983597269265673228582507, −2.11386345682440123814231581710, −0.839114570558405706275007986138,
0.839114570558405706275007986138, 2.11386345682440123814231581710, 2.45281983597269265673228582507, 3.13625436568813031171355036015, 3.82586841758677743731874991009, 4.47020055782771344255608591463, 4.53046448399197474587524858813, 5.70808504593567739297744596633, 6.26317338191743178486844825574, 6.98978401085185422934519850516, 6.99391885917245285038671016395, 8.019809309138138701606525226178, 8.211240608656270383576244000224, 8.493255312495664521477529628378, 8.756988663527870880694039205949, 9.335723948153828938297310453389, 9.624615551705869284953254924198, 10.16781257917926660068681781130, 10.67301765985316947394187938296, 10.97665083703086790483412661559
