| L(s) = 1 | + 2·3-s − 2·5-s + 5·7-s + 9-s + 4·11-s + 2·13-s − 4·15-s − 3·17-s + 6·19-s + 10·21-s − 2·23-s − 9·25-s − 2·27-s − 15·29-s − 10·31-s + 8·33-s − 10·35-s + 5·37-s + 4·39-s − 41-s − 11·43-s − 2·45-s − 16·47-s + 16·49-s − 6·51-s − 2·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.88·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 1.03·15-s − 0.727·17-s + 1.37·19-s + 2.18·21-s − 0.417·23-s − 9/5·25-s − 0.384·27-s − 2.78·29-s − 1.79·31-s + 1.39·33-s − 1.69·35-s + 0.821·37-s + 0.640·39-s − 0.156·41-s − 1.67·43-s − 0.298·45-s − 2.33·47-s + 16/7·49-s − 0.840·51-s − 0.274·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319495389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319495389\) |
| \(L(\frac{3}{2})\) |
|
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 | 2 | | \( 1 \) | |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| good | 5 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.c_n_w_ds |
| 7 | $D_4\times C_2$ | \( 1 - 5 T + 9 T^{2} - 10 T^{3} + 32 T^{4} - 10 p T^{5} + 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.af_j_ak_bg |
| 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 | $D_4\times C_2$ | \( 1 + 3 T - 23 T^{2} - 6 T^{3} + 582 T^{4} - 6 p T^{5} - 23 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.d_ax_ag_wk |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 6 T^{2} + 48 T^{3} - 145 T^{4} + 48 p T^{5} + 6 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_g_bw_afp |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 26 T^{2} - 32 T^{3} + 279 T^{4} - 32 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_aba_abg_kt |
| 29 | $D_4\times C_2$ | \( 1 + 15 T + 115 T^{2} + 780 T^{3} + 4734 T^{4} + 780 p T^{5} + 115 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.p_el_bea_hac |
| 31 | $D_{4}$ | \( ( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.k_dh_xm_gme |
| 37 | $D_4\times C_2$ | \( 1 - 5 T - 51 T^{2} - 10 T^{3} + 3482 T^{4} - 10 p T^{5} - 51 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.af_abz_ak_fdy |
| 41 | $D_4\times C_2$ | \( 1 + T - 77 T^{2} - 4 T^{3} + 4362 T^{4} - 4 p T^{5} - 77 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.41.b_acz_ae_glu |
| 43 | $D_4\times C_2$ | \( 1 + 11 T + 9 T^{2} + 286 T^{3} + 5492 T^{4} + 286 p T^{5} + 9 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.l_j_la_idg |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.q_fs_ccu_sbe |
| 53 | $D_{4}$ | \( ( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.c_hx_ly_xwi |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 38 T^{2} - 256 T^{3} - 1509 T^{4} - 256 p T^{5} - 38 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.e_abm_ajw_acgb |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 195 T^{2} + 1660 T^{3} + 13904 T^{4} + 1660 p T^{5} + 195 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_hn_clw_uou |
| 67 | $D_4\times C_2$ | \( 1 + 3 T - 123 T^{2} - 6 T^{3} + 12332 T^{4} - 6 p T^{5} - 123 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.d_aet_ag_sgi |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 98 T^{2} - 48 T^{3} + 10359 T^{4} - 48 p T^{5} - 98 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.g_adu_abw_pil |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ai_kw_aclk_btkd |
| 79 | $D_{4}$ | \( ( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.g_en_bes_ywy |
| 83 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ho_a_bipi |
| 89 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} + 64 T^{3} - 429 T^{4} + 64 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aq_de_cm_aqn |
| 97 | $D_4\times C_2$ | \( 1 + 3 T - 183 T^{2} - 6 T^{3} + 26582 T^{4} - 6 p T^{5} - 183 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.d_ahb_ag_bnik |
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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
−7.07488795204479524059709318742, −6.96264265913683572612240278835, −6.31168635559563856668240122005, −6.28590305044648220547864289058, −5.97568682716403329538305764059, −5.94949141454711808145175720344, −5.66968542301764447593848301560, −5.25595041379215085643678413242, −5.10872863829497945049402618717, −4.83131496658806996648948481997, −4.54747670145220659170177149446, −4.45650707879461919747528938158, −4.20591705899659565131167298727, −3.72810394727124743937615187666, −3.65392148367691369529338178255, −3.40280375499920224245676397834, −3.29713629071724195185934464338, −3.22144227727043388868327116709, −2.32283185378797734069979987567, −2.23646422044942871730337948384, −1.82250395883263403215168779258, −1.74890429060393229444781550900, −1.55851774112179616209682496415, −1.04097824392170798482442951920, −0.19943707274306792811425025211,
0.19943707274306792811425025211, 1.04097824392170798482442951920, 1.55851774112179616209682496415, 1.74890429060393229444781550900, 1.82250395883263403215168779258, 2.23646422044942871730337948384, 2.32283185378797734069979987567, 3.22144227727043388868327116709, 3.29713629071724195185934464338, 3.40280375499920224245676397834, 3.65392148367691369529338178255, 3.72810394727124743937615187666, 4.20591705899659565131167298727, 4.45650707879461919747528938158, 4.54747670145220659170177149446, 4.83131496658806996648948481997, 5.10872863829497945049402618717, 5.25595041379215085643678413242, 5.66968542301764447593848301560, 5.94949141454711808145175720344, 5.97568682716403329538305764059, 6.28590305044648220547864289058, 6.31168635559563856668240122005, 6.96264265913683572612240278835, 7.07488795204479524059709318742
