| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s − 16-s + 3·19-s + 24-s + 2·25-s − 27-s − 3·38-s + 43-s − 48-s − 2·49-s − 2·50-s + 54-s + 3·57-s − 3·59-s + 64-s + 2·67-s + 2·75-s − 81-s − 86-s − 89-s − 97-s + 2·98-s − 113-s − 3·114-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s − 16-s + 3·19-s + 24-s + 2·25-s − 27-s − 3·38-s + 43-s − 48-s − 2·49-s − 2·50-s + 54-s + 3·57-s − 3·59-s + 64-s + 2·67-s + 2·75-s − 81-s − 86-s − 89-s − 97-s + 2·98-s − 113-s − 3·114-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.158612600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158612600\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 43 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.315098686158124632010179344663, −8.684628793590110106385456550590, −8.312619384126548205201468698656, −7.956043167551944378811023893485, −7.86383806885677918489392953305, −7.28526774715540943733227950291, −7.08047635960023601909025614724, −6.69265774228666734908077081266, −6.10282977578842460019518616237, −5.48529828581590235994796064201, −5.33661142278199766914097043642, −4.71086954363806477268095832404, −4.55332662662374050549504284501, −3.77842558210983361664715463851, −3.38011717631091101524892350941, −2.87308296953977549135333840339, −2.81164942885251415593676373181, −1.84368807712601961423379481072, −1.39430048729816731608275361426, −0.823881671612148118802370586271,
0.823881671612148118802370586271, 1.39430048729816731608275361426, 1.84368807712601961423379481072, 2.81164942885251415593676373181, 2.87308296953977549135333840339, 3.38011717631091101524892350941, 3.77842558210983361664715463851, 4.55332662662374050549504284501, 4.71086954363806477268095832404, 5.33661142278199766914097043642, 5.48529828581590235994796064201, 6.10282977578842460019518616237, 6.69265774228666734908077081266, 7.08047635960023601909025614724, 7.28526774715540943733227950291, 7.86383806885677918489392953305, 7.956043167551944378811023893485, 8.312619384126548205201468698656, 8.684628793590110106385456550590, 9.315098686158124632010179344663
