| L(s) = 1 | + 4·7-s − 11-s + 4·13-s − 2·17-s + 8·19-s + 9·23-s − 3·25-s + 4·29-s + 11·31-s + 7·37-s − 13·41-s − 2·43-s − 11·47-s + 10·49-s − 20·53-s + 2·59-s + 17·61-s − 3·67-s + 8·71-s + 11·73-s − 4·77-s − 27·79-s + 22·83-s − 10·89-s + 16·91-s + 17·97-s + 17·101-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.301·11-s + 1.10·13-s − 0.485·17-s + 1.83·19-s + 1.87·23-s − 3/5·25-s + 0.742·29-s + 1.97·31-s + 1.15·37-s − 2.03·41-s − 0.304·43-s − 1.60·47-s + 10/7·49-s − 2.74·53-s + 0.260·59-s + 2.17·61-s − 0.366·67-s + 0.949·71-s + 1.28·73-s − 0.455·77-s − 3.03·79-s + 2.41·83-s − 1.05·89-s + 1.67·91-s + 1.72·97-s + 1.69·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.14260736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.14260736\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 3 T^{2} - 14 T^{3} + 12 T^{4} - 14 p T^{5} + 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_d_ao_m |
| 11 | $C_2 \wr S_4$ | \( 1 + T + 34 T^{2} + 27 T^{3} + 514 T^{4} + 27 p T^{5} + 34 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.11.b_bi_bb_tu |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 14 T^{2} - 86 T^{3} - 198 T^{4} - 86 p T^{5} + 14 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.c_o_adi_ahq |
| 19 | $C_2 \wr S_4$ | \( 1 - 8 T + 83 T^{2} - 434 T^{3} + 2440 T^{4} - 434 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_df_aqs_dpw |
| 23 | $C_2 \wr S_4$ | \( 1 - 9 T + 91 T^{2} - 500 T^{3} + 3080 T^{4} - 500 p T^{5} + 91 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.aj_dn_atg_eom |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 81 T^{2} - 158 T^{3} + 2824 T^{4} - 158 p T^{5} + 81 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ae_dd_agc_eeq |
| 31 | $C_2 \wr S_4$ | \( 1 - 11 T + 109 T^{2} - 670 T^{3} + 4116 T^{4} - 670 p T^{5} + 109 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.al_ef_azu_gci |
| 37 | $C_2 \wr S_4$ | \( 1 - 7 T + 156 T^{2} - 763 T^{3} + 8794 T^{4} - 763 p T^{5} + 156 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ah_ga_abdj_nag |
| 41 | $C_2 \wr S_4$ | \( 1 + 13 T + 146 T^{2} + 1091 T^{3} + 8010 T^{4} + 1091 p T^{5} + 146 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.n_fq_bpz_lwc |
| 43 | $C_2 \wr S_4$ | \( 1 + 2 T + 93 T^{2} + 374 T^{3} + 4332 T^{4} + 374 p T^{5} + 93 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.c_dp_ok_gkq |
| 47 | $C_2 \wr S_4$ | \( 1 + 11 T + 173 T^{2} + 1198 T^{3} + 11124 T^{4} + 1198 p T^{5} + 173 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.l_gr_buc_qlw |
| 53 | $C_2 \wr S_4$ | \( 1 + 20 T + 321 T^{2} + 3158 T^{3} + 27408 T^{4} + 3158 p T^{5} + 321 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.u_mj_erm_booe |
| 59 | $C_2 \wr S_4$ | \( 1 - 2 T + 112 T^{2} - 26 T^{3} + 8398 T^{4} - 26 p T^{5} + 112 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ac_ei_aba_mla |
| 61 | $C_2 \wr S_4$ | \( 1 - 17 T + 198 T^{2} - 1587 T^{3} + 13650 T^{4} - 1587 p T^{5} + 198 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ar_hq_acjb_ufa |
| 67 | $C_2 \wr S_4$ | \( 1 + 3 T + 208 T^{2} + 703 T^{3} + 18910 T^{4} + 703 p T^{5} + 208 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.d_ia_bbb_bbzi |
| 71 | $C_2 \wr S_4$ | \( 1 - 8 T + 154 T^{2} - 1904 T^{3} + 12362 T^{4} - 1904 p T^{5} + 154 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ai_fy_acvg_shm |
| 73 | $C_2 \wr S_4$ | \( 1 - 11 T + 271 T^{2} - 2296 T^{3} + 29094 T^{4} - 2296 p T^{5} + 271 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.al_kl_adki_brba |
| 79 | $C_2 \wr S_4$ | \( 1 + 27 T + 491 T^{2} + 6316 T^{3} + 65104 T^{4} + 6316 p T^{5} + 491 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.bb_sx_jiy_dsia |
| 83 | $C_2 \wr S_4$ | \( 1 - 22 T + 447 T^{2} - 5510 T^{3} + 59992 T^{4} - 5510 p T^{5} + 447 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.aw_rf_aidy_dktk |
| 89 | $C_2 \wr S_4$ | \( 1 + 10 T + 163 T^{2} + 1174 T^{3} + 10696 T^{4} + 1174 p T^{5} + 163 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.k_gh_bte_pvk |
| 97 | $C_2 \wr S_4$ | \( 1 - 17 T + 335 T^{2} - 4172 T^{3} + 47266 T^{4} - 4172 p T^{5} + 335 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ar_mx_agem_crxy |
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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
−5.66051773139312022236227176632, −5.31256781033826296966953699174, −5.13283387709862457576103981033, −5.02426030661892744984341084782, −4.86762105284003440103693240873, −4.83561024090134437810088592258, −4.56843735438606604467318465215, −4.40856275268268449109123355009, −4.14044972827300390241985008862, −3.87583716741780719841541600639, −3.69259720961578281057813264690, −3.41363891296842124354222988500, −3.36389886891402316830682116940, −2.89479879455554996803438842354, −2.89419216713637773271397663655, −2.79950348906534091346170360713, −2.60438632312997737210628712451, −1.84405319083989552535471389484, −1.80919492917553692367072502502, −1.74097654536974917123883100715, −1.72174038740676777156105425834, −1.04856556188401858796140759272, −0.859246263859177735863788214517, −0.67705368413546290688562388212, −0.51867314823587206879646350188,
0.51867314823587206879646350188, 0.67705368413546290688562388212, 0.859246263859177735863788214517, 1.04856556188401858796140759272, 1.72174038740676777156105425834, 1.74097654536974917123883100715, 1.80919492917553692367072502502, 1.84405319083989552535471389484, 2.60438632312997737210628712451, 2.79950348906534091346170360713, 2.89419216713637773271397663655, 2.89479879455554996803438842354, 3.36389886891402316830682116940, 3.41363891296842124354222988500, 3.69259720961578281057813264690, 3.87583716741780719841541600639, 4.14044972827300390241985008862, 4.40856275268268449109123355009, 4.56843735438606604467318465215, 4.83561024090134437810088592258, 4.86762105284003440103693240873, 5.02426030661892744984341084782, 5.13283387709862457576103981033, 5.31256781033826296966953699174, 5.66051773139312022236227176632
