| L(s) = 1 | − 2-s + 4·3-s − 4·6-s + 10·9-s + 8·11-s + 7·13-s + 16-s + 6·17-s − 10·18-s + 3·19-s − 8·22-s − 2·23-s − 7·26-s + 20·27-s + 16·29-s + 9·31-s − 3·32-s + 32·33-s − 6·34-s − 8·37-s − 3·38-s + 28·39-s + 4·41-s − 5·43-s + 2·46-s − 6·47-s + 4·48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s − 1.63·6-s + 10/3·9-s + 2.41·11-s + 1.94·13-s + 1/4·16-s + 1.45·17-s − 2.35·18-s + 0.688·19-s − 1.70·22-s − 0.417·23-s − 1.37·26-s + 3.84·27-s + 2.97·29-s + 1.61·31-s − 0.530·32-s + 5.57·33-s − 1.02·34-s − 1.31·37-s − 0.486·38-s + 4.48·39-s + 0.624·41-s − 0.762·43-s + 0.294·46-s − 0.875·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(26.86469128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.86469128\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 5 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + T^{2} + T^{3} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.2.b_b_b_a |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 36 T^{2} - 84 T^{3} + 222 T^{4} - 84 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ai_bk_adg_io |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 33 T^{2} - 94 T^{3} + 326 T^{4} - 94 p T^{5} + 33 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ah_bh_adq_mo |
| 17 | $C_2 \wr S_4$ | \( 1 - 6 T + 48 T^{2} - 198 T^{3} + 1094 T^{4} - 198 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_bw_ahq_bqc |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 39 T^{2} + 4 T^{3} + 564 T^{4} + 4 p T^{5} + 39 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ad_bn_e_vs |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 24 T^{2} + 66 T^{3} + 1006 T^{4} + 66 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_y_co_bms |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.29.aq_ie_aclk_qcc |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 68 T^{2} - 261 T^{3} + 1526 T^{4} - 261 p T^{5} + 68 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.aj_cq_akb_cgs |
| 37 | $C_2 \wr S_4$ | \( 1 + 8 T + 86 T^{2} + 744 T^{3} + 4399 T^{4} + 744 p T^{5} + 86 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.i_di_bcq_gnf |
| 41 | $C_2 \wr S_4$ | \( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 256 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ae_ea_ajw_hja |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 120 T^{2} + 677 T^{3} + 6782 T^{4} + 677 p T^{5} + 120 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.f_eq_bab_kaw |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T + 124 T^{2} + 346 T^{3} + 6382 T^{4} + 346 p T^{5} + 124 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.g_eu_ni_jlm |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T + 136 T^{2} - 662 T^{3} + 9910 T^{4} - 662 p T^{5} + 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ag_fg_azm_ore |
| 59 | $C_2 \wr S_4$ | \( 1 - 10 T + 196 T^{2} - 1726 T^{3} + 16158 T^{4} - 1726 p T^{5} + 196 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ak_ho_acok_xxm |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.61.u_pe_gea_cjap |
| 67 | $C_2 \wr S_4$ | \( 1 + T + 251 T^{2} + 192 T^{3} + 24716 T^{4} + 192 p T^{5} + 251 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.67.b_jr_hk_bkoq |
| 71 | $C_2 \wr S_4$ | \( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 1806 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aw_js_acrm_tes |
| 73 | $C_2 \wr S_4$ | \( 1 + 4 T + 170 T^{2} - 48 T^{3} + 12595 T^{4} - 48 p T^{5} + 170 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.e_go_abw_sql |
| 79 | $C_2 \wr S_4$ | \( 1 - 8 T + 246 T^{2} - 1764 T^{3} + 26539 T^{4} - 1764 p T^{5} + 246 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ai_jm_acpw_bngt |
| 83 | $C_2 \wr S_4$ | \( 1 + 2 T + 220 T^{2} + 894 T^{3} + 22430 T^{4} + 894 p T^{5} + 220 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.c_im_bik_bhes |
| 89 | $C_2 \wr S_4$ | \( 1 + 10 T + 360 T^{2} + 2586 T^{3} + 48262 T^{4} + 2586 p T^{5} + 360 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.k_nw_dvm_ctkg |
| 97 | $C_2 \wr S_4$ | \( 1 - 12 T + 330 T^{2} - 3296 T^{3} + 45123 T^{4} - 3296 p T^{5} + 330 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.am_ms_aewu_cotn |
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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
−6.18286467821691173842926500918, −6.14102386796058597364996562557, −5.50660298487745357579122666579, −5.45592868743314848570689061454, −5.39795563954793688890297833304, −4.76498351941048844350477264440, −4.66554663858442666284954550972, −4.53743068132376403411870763756, −4.50518399095589251940270800036, −4.02439322017929864437314713636, −3.84748256780947505613342800496, −3.67156994569098093705357822631, −3.43759908369699768680896824807, −3.36954725122701693280640375883, −3.15968747201112293133292129305, −3.04295466575860592242375574755, −2.71055993464176665232134085869, −2.33966052127041127078406232025, −2.13003809535980621775839909188, −1.72137177009089446718328704392, −1.60646624997846904020052753148, −1.40158335348539744536863209348, −0.890310623491680992123692546607, −0.816446562878070847736298944537, −0.812057415163307660590584085950,
0.812057415163307660590584085950, 0.816446562878070847736298944537, 0.890310623491680992123692546607, 1.40158335348539744536863209348, 1.60646624997846904020052753148, 1.72137177009089446718328704392, 2.13003809535980621775839909188, 2.33966052127041127078406232025, 2.71055993464176665232134085869, 3.04295466575860592242375574755, 3.15968747201112293133292129305, 3.36954725122701693280640375883, 3.43759908369699768680896824807, 3.67156994569098093705357822631, 3.84748256780947505613342800496, 4.02439322017929864437314713636, 4.50518399095589251940270800036, 4.53743068132376403411870763756, 4.66554663858442666284954550972, 4.76498351941048844350477264440, 5.39795563954793688890297833304, 5.45592868743314848570689061454, 5.50660298487745357579122666579, 6.14102386796058597364996562557, 6.18286467821691173842926500918
