| L(s) = 1 | + 5·2-s − 9·3-s + 8·4-s + 19·5-s − 45·6-s + 22·7-s − 5·8-s + 54·9-s + 95·10-s − 16·11-s − 72·12-s + 89·13-s + 110·14-s − 171·15-s − 25·16-s − 131·17-s + 270·18-s + 93·19-s + 152·20-s − 198·21-s − 80·22-s + 138·23-s + 45·24-s + 125·25-s + 445·26-s − 243·27-s + 176·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.73·3-s + 4-s + 1.69·5-s − 3.06·6-s + 1.18·7-s − 0.220·8-s + 2·9-s + 3.00·10-s − 0.438·11-s − 1.73·12-s + 1.89·13-s + 2.09·14-s − 2.94·15-s − 0.390·16-s − 1.86·17-s + 3.53·18-s + 1.12·19-s + 1.69·20-s − 2.05·21-s − 0.775·22-s + 1.25·23-s + 0.382·24-s + 25-s + 3.35·26-s − 1.73·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.533686096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.533686096\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 13 | $C_2$ | \( 1 - 89 T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 19 T + 236 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16 T - 1075 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 131 T + 12248 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 93 T + 1790 T^{2} - 93 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 p T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 86 T - 16993 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 173 T + 138 T^{2} - 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 433 T + p^{3} T^{2} )( 1 + 110 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 69 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 369 T + 32338 T^{2} + 369 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 306 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 420 T - 28979 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 383 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 139 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 237 T - 301742 T^{2} + 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 518 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 41 T - 491358 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 905 T + 247238 T^{2} - 905 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 929 T + 158072 T^{2} - 929 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p^{3} T^{2} )^{2} \) |
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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
−14.02207568520840215378439088428, −13.07313953120447257439807382040, −12.60185014128964981424957385202, −11.54201868476541224009115811232, −11.48908193636433939415423736825, −10.83781678622789635124847777719, −10.69670866503310099177834666311, −9.667130061559705103053551657867, −9.310382993033514115339525414673, −8.465660144792896177762803982820, −7.64762381401558634090657408122, −6.59339433084505640016949536693, −6.19733014740808804591266001620, −5.90135868179606925328405833447, −5.11396565235890119933884197765, −4.89109709874297300830467142229, −4.40986171271313169915020271398, −3.32144821236275162403280548928, −1.94240714248798085821633477947, −1.08994227204926132498959958800,
1.08994227204926132498959958800, 1.94240714248798085821633477947, 3.32144821236275162403280548928, 4.40986171271313169915020271398, 4.89109709874297300830467142229, 5.11396565235890119933884197765, 5.90135868179606925328405833447, 6.19733014740808804591266001620, 6.59339433084505640016949536693, 7.64762381401558634090657408122, 8.465660144792896177762803982820, 9.310382993033514115339525414673, 9.667130061559705103053551657867, 10.69670866503310099177834666311, 10.83781678622789635124847777719, 11.48908193636433939415423736825, 11.54201868476541224009115811232, 12.60185014128964981424957385202, 13.07313953120447257439807382040, 14.02207568520840215378439088428
