| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 2·7-s + 10-s − 2·14-s + 15-s + 2·17-s − 2·21-s − 30-s − 31-s + 32-s − 2·34-s − 2·35-s − 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s − 61-s + 62-s − 64-s − 67-s + 2·70-s − 73-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 2·7-s + 10-s − 2·14-s + 15-s + 2·17-s − 2·21-s − 30-s − 31-s + 32-s − 2·34-s − 2·35-s − 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s − 61-s + 62-s − 64-s − 67-s + 2·70-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1603281878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1603281878\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−14.04573251757204190644351240073, −13.65194338558675342126772884310, −12.68729914443010867183501235312, −12.04645147468382634720510609770, −11.84702716241586137927181515730, −11.44893583765498948948023432146, −10.99785240360884651423191253530, −10.44337523099519291188430246284, −9.878008452330803346119504481607, −9.197151922943741932582102142604, −8.439959401615032609354256443612, −8.252672509741494813962148682655, −7.59622314204297360302731343397, −7.30241031392590951611595560944, −6.12094459230689769919386566111, −5.48989736619258142965789236319, −4.97597257629778139191185706419, −4.26497223589162521783933658687, −3.27448017805867294444555410081, −1.52402899467920246594218058938,
1.52402899467920246594218058938, 3.27448017805867294444555410081, 4.26497223589162521783933658687, 4.97597257629778139191185706419, 5.48989736619258142965789236319, 6.12094459230689769919386566111, 7.30241031392590951611595560944, 7.59622314204297360302731343397, 8.252672509741494813962148682655, 8.439959401615032609354256443612, 9.197151922943741932582102142604, 9.878008452330803346119504481607, 10.44337523099519291188430246284, 10.99785240360884651423191253530, 11.44893583765498948948023432146, 11.84702716241586137927181515730, 12.04645147468382634720510609770, 12.68729914443010867183501235312, 13.65194338558675342126772884310, 14.04573251757204190644351240073
