| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 3·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s + 6·10-s + 11-s + 6·12-s + 2·13-s − 4·14-s + 6·15-s + 5·16-s − 17-s + 6·18-s + 3·19-s + 9·20-s − 4·21-s + 2·22-s − 7·23-s + 8·24-s + 25-s + 4·26-s + 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.34·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 1.89·10-s + 0.301·11-s + 1.73·12-s + 0.554·13-s − 1.06·14-s + 1.54·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s + 0.688·19-s + 2.01·20-s − 0.872·21-s + 0.426·22-s − 1.45·23-s + 1.63·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.066534739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.066534739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 144 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 26 T + 318 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.96102185939495039898716611437, −10.41857078250021618048528429289, −10.22060041031839152001460924305, −9.756888510714417580519514037867, −9.233370560023010947145845166715, −9.010485694780640031278337377431, −8.305995732921860230626379067102, −7.83825761795388458926986676872, −7.32618673380141590440426968804, −6.71997963436205347221010032644, −6.32928801133355306306372815819, −6.09386238561401144317148233861, −5.20605424187117081970844569536, −5.20040415974629017633015291518, −4.13517339401693938788755970479, −3.73522433295182470573736754474, −3.32218683883157689035440528794, −2.67685834892058680739712973755, −1.96532787879004213571976830321, −1.61386041996840783631232130153,
1.61386041996840783631232130153, 1.96532787879004213571976830321, 2.67685834892058680739712973755, 3.32218683883157689035440528794, 3.73522433295182470573736754474, 4.13517339401693938788755970479, 5.20040415974629017633015291518, 5.20605424187117081970844569536, 6.09386238561401144317148233861, 6.32928801133355306306372815819, 6.71997963436205347221010032644, 7.32618673380141590440426968804, 7.83825761795388458926986676872, 8.305995732921860230626379067102, 9.010485694780640031278337377431, 9.233370560023010947145845166715, 9.756888510714417580519514037867, 10.22060041031839152001460924305, 10.41857078250021618048528429289, 10.96102185939495039898716611437
