| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 2·8-s + 9-s + 4·11-s − 6·12-s − 4·17-s + 2·18-s + 8·22-s + 8·23-s − 4·24-s − 4·25-s + 2·27-s − 4·29-s − 16·31-s − 6·32-s − 8·33-s − 8·34-s + 3·36-s + 4·37-s − 16·41-s − 8·43-s + 12·44-s + 16·46-s − 24·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.707·8-s + 1/3·9-s + 1.20·11-s − 1.73·12-s − 0.970·17-s + 0.471·18-s + 1.70·22-s + 1.66·23-s − 0.816·24-s − 4/5·25-s + 0.384·27-s − 0.742·29-s − 2.87·31-s − 1.06·32-s − 1.39·33-s − 1.37·34-s + 1/2·36-s + 0.657·37-s − 2.49·41-s − 1.21·43-s + 1.80·44-s + 2.35·46-s − 3.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6770926459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6770926459\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_b_c_ad |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_e_a_cc |
| 7 | $C_2^3$ | \( 1 - 6 T^{2} - 13 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_ag_a_an |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ae_ak_aq_op |
| 17 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_k_aei_auf |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_abe_a_ut |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_c_aey_csx |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.e_abu_q_dtr |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.q_hw_cke_pza |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 30 T^{2} + 112 T^{3} + 155 T^{4} + 112 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_abe_ei_fz |
| 41 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} + 896 T^{3} + 6867 T^{4} + 896 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.q_eo_bim_ked |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} - 128 T^{3} + 299 T^{4} - 128 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.i_ag_aey_ln |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.y_nc_fdw_boty |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.53.i_jc_bye_bcss |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 74 T^{2} - 112 T^{3} + 3675 T^{4} - 112 p T^{5} - 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.e_acw_aei_flj |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 496 T^{3} - 4693 T^{4} - 496 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.e_s_atc_agyn |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} + 64 T^{3} + 11387 T^{4} + 64 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.i_ada_cm_qvz |
| 71 | $C_2^2$ | \( ( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.e_afa_q_wkh |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ay_rc_ahxw_dcdy |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ci_a_tus |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.i_lg_cqa_camo |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 262 T^{2} + 3264 T^{3} + 39411 T^{4} + 3264 p T^{5} + 262 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.y_kc_evo_cghv |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 150 T^{2} + 112 T^{3} + 16595 T^{4} + 112 p T^{5} - 150 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ae_afu_ei_yoh |
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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
−7.84922139129924135166133311769, −7.39463542524467869950650409295, −7.23415362389483094214672477576, −7.12504765337616079950131112950, −6.53573908566459362288316204647, −6.53482904977824713737872649975, −6.45426770035027835327806725168, −6.38003513046821651779530456428, −5.74418499667077562132288631447, −5.56983670656069765919798782972, −5.50473822978601613322554712979, −5.02599982122258267767009437572, −4.96605272585828715322306163828, −4.72640259683560524811398165544, −4.45577918369397104734380978904, −3.79536080434194564220302980263, −3.71828732660315513116485539770, −3.71455226739373052324639476623, −3.05442107698060846600322638950, −3.03746318972948242340605674710, −2.55457321727063998278556942297, −1.74652173092766292815956757184, −1.68544522410388219566703565469, −1.55583427990452834321666908859, −0.20936790227373933743897108075,
0.20936790227373933743897108075, 1.55583427990452834321666908859, 1.68544522410388219566703565469, 1.74652173092766292815956757184, 2.55457321727063998278556942297, 3.03746318972948242340605674710, 3.05442107698060846600322638950, 3.71455226739373052324639476623, 3.71828732660315513116485539770, 3.79536080434194564220302980263, 4.45577918369397104734380978904, 4.72640259683560524811398165544, 4.96605272585828715322306163828, 5.02599982122258267767009437572, 5.50473822978601613322554712979, 5.56983670656069765919798782972, 5.74418499667077562132288631447, 6.38003513046821651779530456428, 6.45426770035027835327806725168, 6.53482904977824713737872649975, 6.53573908566459362288316204647, 7.12504765337616079950131112950, 7.23415362389483094214672477576, 7.39463542524467869950650409295, 7.84922139129924135166133311769
