| L(s) = 1 | − 3-s + 2·4-s + 5-s + 3·9-s + 3·11-s − 2·12-s + 10·13-s − 15-s − 3·17-s − 2·19-s + 2·20-s + 6·23-s − 8·27-s + 6·29-s + 4·31-s − 3·33-s + 6·36-s − 2·37-s − 10·39-s − 24·41-s − 20·43-s + 6·44-s + 3·45-s − 9·47-s + 3·51-s + 20·52-s − 12·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 0.447·5-s + 9-s + 0.904·11-s − 0.577·12-s + 2.77·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s + 0.447·20-s + 1.25·23-s − 1.53·27-s + 1.11·29-s + 0.718·31-s − 0.522·33-s + 36-s − 0.328·37-s − 1.60·39-s − 3.74·41-s − 3.04·43-s + 0.904·44-s + 0.447·45-s − 1.31·47-s + 0.420·51-s + 2.77·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.905753381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905753381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−12.10194804678235806874042783220, −11.65727737368445366670696295834, −11.45004176964186420458997400265, −10.99732400934233756383588807829, −10.44868144079610207187074374545, −10.23014148645109283484995464902, −9.440478056710243554149160591124, −8.967661317560419698394995409895, −8.291621576686281165493424515429, −8.173870342928776447521522365671, −6.79828732350708024546049650676, −6.78827030942083609426156456071, −6.45160428522022824140524885980, −6.03821227235137514661533204502, −4.99431691664906587192565670923, −4.65935311714706110465415933365, −3.48941768776716897594935397528, −3.37297023196331269267382799009, −1.64946938923012272135865661838, −1.60588733443480794334045228149,
1.60588733443480794334045228149, 1.64946938923012272135865661838, 3.37297023196331269267382799009, 3.48941768776716897594935397528, 4.65935311714706110465415933365, 4.99431691664906587192565670923, 6.03821227235137514661533204502, 6.45160428522022824140524885980, 6.78827030942083609426156456071, 6.79828732350708024546049650676, 8.173870342928776447521522365671, 8.291621576686281165493424515429, 8.967661317560419698394995409895, 9.440478056710243554149160591124, 10.23014148645109283484995464902, 10.44868144079610207187074374545, 10.99732400934233756383588807829, 11.45004176964186420458997400265, 11.65727737368445366670696295834, 12.10194804678235806874042783220
