| L(s) = 1 | − 3·3-s + 5-s + 2·9-s + 4·11-s + 8·13-s − 3·15-s − 4·17-s − 14·19-s + 8·23-s − 4·25-s + 2·27-s + 4·29-s − 5·31-s − 12·33-s − 4·37-s − 24·39-s + 7·41-s + 19·43-s + 2·45-s − 8·47-s + 12·51-s + 5·53-s + 4·55-s + 42·57-s + 23·61-s + 8·65-s + 15·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2/3·9-s + 1.20·11-s + 2.21·13-s − 0.774·15-s − 0.970·17-s − 3.21·19-s + 1.66·23-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 0.898·31-s − 2.08·33-s − 0.657·37-s − 3.84·39-s + 1.09·41-s + 2.89·43-s + 0.298·45-s − 1.16·47-s + 1.68·51-s + 0.686·53-s + 0.539·55-s + 5.56·57-s + 2.94·61-s + 0.992·65-s + 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.284821203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.284821203\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + p T + 7 T^{2} + 13 T^{3} + 28 T^{4} + 13 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.3.d_h_n_bc |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + p T^{2} + 7 T^{3} + 4 T^{4} + 7 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ab_f_h_e |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ae_bo_aem_ze |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ai_bk_ahc_bgw |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 116 T^{2} + 710 T^{3} + 3510 T^{4} + 710 p T^{5} + 116 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.o_em_bbi_ffa |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 72 T^{2} - 328 T^{3} + 1998 T^{4} - 328 p T^{5} + 72 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ai_cu_amq_cyw |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 48 T^{2} - 364 T^{3} + 1118 T^{4} - 364 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ae_bw_aoa_bra |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 5 T + 39 T^{2} - 35 T^{3} + 96 T^{4} - 35 p T^{5} + 39 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.f_bn_abj_ds |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 80 T^{2} + 140 T^{3} + 2878 T^{4} + 140 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_dc_fk_egs |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 99 T^{2} - 625 T^{3} + 5720 T^{4} - 625 p T^{5} + 99 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ah_dv_ayb_ima |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 19 T + 237 T^{2} - 2183 T^{3} + 16508 T^{4} - 2183 p T^{5} + 237 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.at_jd_adfz_yky |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 76 T^{2} + 616 T^{3} + 5542 T^{4} + 616 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.i_cy_xs_ife |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 111 T^{2} - 575 T^{3} + 6632 T^{4} - 575 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.af_eh_awd_jvc |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 192 T^{2} - 80 T^{3} + 15758 T^{4} - 80 p T^{5} + 192 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_hk_adc_xic |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 23 T + 357 T^{2} - 3847 T^{3} + 34148 T^{4} - 3847 p T^{5} + 357 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ax_nt_afrz_bynk |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 239 T^{2} - 2555 T^{3} + 22872 T^{4} - 2555 p T^{5} + 239 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ap_jf_aduh_bhvs |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 152 T^{2} - 1018 T^{3} + 10798 T^{4} - 1018 p T^{5} + 152 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ac_fw_abne_pzi |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 91 T^{2} - 155 T^{3} + 2872 T^{4} - 155 p T^{5} + 91 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.af_dn_afz_egm |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 396 T^{2} + 4920 T^{3} + 49830 T^{4} + 4920 p T^{5} + 396 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.y_pg_hhg_cvso |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 212 T^{2} + 1890 T^{3} + 25814 T^{4} + 1890 p T^{5} + 212 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.k_ie_cus_bmew |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 136 T^{2} - 1280 T^{3} + 16110 T^{4} - 1280 p T^{5} + 136 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aq_fg_abxg_xvq |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 323 T^{2} - 2665 T^{3} + 37944 T^{4} - 2665 p T^{5} + 323 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ap_ml_adyn_cedk |
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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
−5.76096260248053914842652930339, −5.42615384249961732219212554446, −5.25470854676165688603108532039, −5.15544365015141235363024693265, −5.14424781965367839078751091022, −4.59912396813184519073771555395, −4.30572362180233726958254915322, −4.26801512884532504390371906827, −4.18174170065592305399091753148, −3.94764202151371512203018688562, −3.90808371430340894615309799589, −3.64209072641005919856698237469, −3.41956828343951904900963304789, −2.98078345062606687215463218320, −2.69949463268583087126774750087, −2.67038971433166155671059772570, −2.48407737195132234471636638810, −1.96646244807072514841848332577, −1.94851637693061909523497196218, −1.65893852111051538172853352332, −1.40035440216175894486078067914, −1.14899885655113674606830477527, −0.64136529261049460291805382197, −0.57926904414511629064688258635, −0.37923500615821965196408354497,
0.37923500615821965196408354497, 0.57926904414511629064688258635, 0.64136529261049460291805382197, 1.14899885655113674606830477527, 1.40035440216175894486078067914, 1.65893852111051538172853352332, 1.94851637693061909523497196218, 1.96646244807072514841848332577, 2.48407737195132234471636638810, 2.67038971433166155671059772570, 2.69949463268583087126774750087, 2.98078345062606687215463218320, 3.41956828343951904900963304789, 3.64209072641005919856698237469, 3.90808371430340894615309799589, 3.94764202151371512203018688562, 4.18174170065592305399091753148, 4.26801512884532504390371906827, 4.30572362180233726958254915322, 4.59912396813184519073771555395, 5.14424781965367839078751091022, 5.15544365015141235363024693265, 5.25470854676165688603108532039, 5.42615384249961732219212554446, 5.76096260248053914842652930339
