| L(s) = 1 | + 4-s − 8·13-s + 4·16-s − 12·17-s − 18·23-s + 13·25-s + 6·29-s + 8·43-s + 49-s − 8·52-s + 24·53-s + 2·61-s + 11·64-s − 12·68-s − 4·79-s − 18·92-s + 13·100-s + 12·101-s + 26·103-s + 6·107-s − 12·113-s + 6·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.21·13-s + 16-s − 2.91·17-s − 3.75·23-s + 13/5·25-s + 1.11·29-s + 1.21·43-s + 1/7·49-s − 1.10·52-s + 3.29·53-s + 0.256·61-s + 11/8·64-s − 1.45·68-s − 0.450·79-s − 1.87·92-s + 1.29·100-s + 1.19·101-s + 2.56·103-s + 0.580·107-s − 1.12·113-s + 0.557·116-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706033467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706033467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_ab_a_ad |
| 5 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 81 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_an_a_dd |
| 7 | $D_4\times C_2$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_ab_a_ad |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + 141 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ab_a_fl |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.17.m_es_bbs_fkl |
| 19 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_o_a_bdr |
| 23 | $D_{4}$ | \( ( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.s_hj_cca_loj |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 49 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ag_ed_asa_gvd |
| 31 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 4173 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adt_a_gen |
| 37 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3633 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_adh_a_fjt |
| 41 | $C_4\times C_2$ | \( 1 - 61 T^{2} + 1761 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acj_a_cpt |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ai_ec_abbc_knb |
| 47 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7923 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_afa_a_lst |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.53.ay_qm_agya_ciss |
| 59 | $D_4\times C_2$ | \( 1 - 58 T^{2} + 4923 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_acg_a_hhj |
| 61 | $D_{4}$ | \( ( 1 - T + 111 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ac_ip_ang_bdkv |
| 67 | $D_4\times C_2$ | \( 1 - 133 T^{2} + 10869 T^{4} - 133 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_afd_a_qcb |
| 71 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 19563 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ahu_a_bcyl |
| 73 | $C_2^2$ | \( ( 1 - 101 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ahu_a_bewh |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.79.e_mk_bkq_ceut |
| 83 | $D_4\times C_2$ | \( 1 - 274 T^{2} + 31827 T^{4} - 274 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ako_a_bvcd |
| 89 | $D_4\times C_2$ | \( 1 - 241 T^{2} + 30081 T^{4} - 241 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ajh_a_bsmz |
| 97 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 36798 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_aku_a_ccli |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.15404742211594191940294453331, −6.88266494470479195649624525152, −6.75758066529618214472557046375, −6.48215948075015588449771928903, −6.17693665638201299434087223107, −5.88032173625355434476014638914, −5.87280222293752893668015956409, −5.70760020728803307378866326988, −5.10538344993652355935536789997, −4.99208286788931353351411261474, −4.76249051121958318609130198510, −4.59314341872781570898856495885, −4.41185230553677473591920755164, −4.02116971063467184599825523445, −3.79475746571260827112379773086, −3.64509898056970800350878528431, −3.25591036888317828040292066308, −2.56590474484180694495779948606, −2.51730496529652910928778603663, −2.47089694927021293220025756260, −2.29194669236751637241015422442, −1.85439072208491495839990728983, −1.38703611336561299415088507410, −0.75651053382172763557699393223, −0.35054361063876387162813206069,
0.35054361063876387162813206069, 0.75651053382172763557699393223, 1.38703611336561299415088507410, 1.85439072208491495839990728983, 2.29194669236751637241015422442, 2.47089694927021293220025756260, 2.51730496529652910928778603663, 2.56590474484180694495779948606, 3.25591036888317828040292066308, 3.64509898056970800350878528431, 3.79475746571260827112379773086, 4.02116971063467184599825523445, 4.41185230553677473591920755164, 4.59314341872781570898856495885, 4.76249051121958318609130198510, 4.99208286788931353351411261474, 5.10538344993652355935536789997, 5.70760020728803307378866326988, 5.87280222293752893668015956409, 5.88032173625355434476014638914, 6.17693665638201299434087223107, 6.48215948075015588449771928903, 6.75758066529618214472557046375, 6.88266494470479195649624525152, 7.15404742211594191940294453331
