| L(s) = 1 | + 4·3-s − 2·5-s − 2·7-s + 10·9-s + 4·11-s + 4·13-s − 8·15-s − 4·17-s − 10·19-s − 8·21-s − 6·25-s + 20·27-s − 28·29-s − 2·31-s + 16·33-s + 4·35-s + 16·39-s − 2·41-s − 12·43-s − 20·45-s − 4·47-s − 14·49-s − 16·51-s + 8·53-s − 8·55-s − 40·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s − 0.755·7-s + 10/3·9-s + 1.20·11-s + 1.10·13-s − 2.06·15-s − 0.970·17-s − 2.29·19-s − 1.74·21-s − 6/5·25-s + 3.84·27-s − 5.19·29-s − 0.359·31-s + 2.78·33-s + 0.676·35-s + 2.56·39-s − 0.312·41-s − 1.82·43-s − 2.98·45-s − 0.583·47-s − 2·49-s − 2.24·51-s + 1.09·53-s − 1.07·55-s − 5.29·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 2 T + 2 p T^{2} + 34 T^{3} + 54 T^{4} + 34 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.c_k_bi_cc |
| 7 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 2 T + 18 T^{2} + 46 T^{3} + 158 T^{4} + 46 p T^{5} + 18 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_s_bu_gc |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ae_u_acy_pe |
| 17 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 4 T + 40 T^{2} + 140 T^{3} + 798 T^{4} + 140 p T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_bo_fk_bes |
| 19 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 10 T + 82 T^{2} + 430 T^{3} + 2158 T^{4} + 430 p T^{5} + 82 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.k_de_qo_dfa |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ca_a_cos |
| 29 | $D_{4}$ | \( ( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.bc_pk_flc_bisc |
| 31 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 2 T + 114 T^{2} + 190 T^{3} + 5150 T^{4} + 190 p T^{5} + 114 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.c_ek_hi_hqc |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ee_a_ijm |
| 41 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 2 T + 34 T^{2} + 250 T^{3} + 2550 T^{4} + 250 p T^{5} + 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.c_bi_jq_duc |
| 43 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 12 T + 192 T^{2} + 1452 T^{3} + 12638 T^{4} + 1452 p T^{5} + 192 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.m_hk_cdw_ssc |
| 47 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 4 T + 124 T^{2} + 428 T^{3} + 7194 T^{4} + 428 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.e_eu_qm_kqs |
| 53 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 - 8 T + 52 T^{2} + 392 T^{3} - 2922 T^{4} + 392 p T^{5} + 52 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ai_ca_pc_aeik |
| 59 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 - 8 T + 164 T^{2} - 1064 T^{3} + 13786 T^{4} - 1064 p T^{5} + 164 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ai_gi_aboy_ukg |
| 61 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 20 T + 240 T^{2} + 2140 T^{3} + 17822 T^{4} + 2140 p T^{5} + 240 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_jg_dei_bajm |
| 67 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 - 14 T + 210 T^{2} - 1930 T^{3} + 20318 T^{4} - 1930 p T^{5} + 210 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ao_ic_acwg_bebm |
| 71 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 12426 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_eu_a_sjy |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.73.i_me_cqq_cane |
| 79 | $D_4\times C_2$ | \( 1 + 132 T^{2} + 12998 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_fc_a_tfy |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 152 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ae_lw_abke_cdoc |
| 89 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 - 2 T + 106 T^{2} + 542 T^{3} + 5190 T^{4} + 542 p T^{5} + 106 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ac_ec_uw_hrq |
| 97 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 + 12 T + 108 T^{2} - 204 T^{3} - 922 T^{4} - 204 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.m_ee_ahw_abjm |
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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
−5.95390210477292352580192271917, −5.47592264342608419892298603881, −5.34152831180751455699556598978, −5.13327127745297255787187693439, −4.94367341308322208837771949176, −4.71996473187141964986399992945, −4.37336678769706453145592219088, −4.33259596293952885276479915513, −4.05827725822481862385219946643, −3.98516664407078837645696368959, −3.82186173440795356379042692314, −3.62991794402978933686522942203, −3.59886979005275389246280568174, −3.45916054150928954340289829081, −3.20649657804602254041054218001, −3.09233912456858787102882763045, −2.68392738963755480633573887328, −2.28353038316702320841018945193, −2.24147569755966021836320918550, −2.22062174195982865630868913820, −1.93611721327050846225017580954, −1.63303170338800049630752387088, −1.49568302147300845846951279410, −1.24652064911781083354854438495, −1.11674969933323348641381321519, 0, 0, 0, 0,
1.11674969933323348641381321519, 1.24652064911781083354854438495, 1.49568302147300845846951279410, 1.63303170338800049630752387088, 1.93611721327050846225017580954, 2.22062174195982865630868913820, 2.24147569755966021836320918550, 2.28353038316702320841018945193, 2.68392738963755480633573887328, 3.09233912456858787102882763045, 3.20649657804602254041054218001, 3.45916054150928954340289829081, 3.59886979005275389246280568174, 3.62991794402978933686522942203, 3.82186173440795356379042692314, 3.98516664407078837645696368959, 4.05827725822481862385219946643, 4.33259596293952885276479915513, 4.37336678769706453145592219088, 4.71996473187141964986399992945, 4.94367341308322208837771949176, 5.13327127745297255787187693439, 5.34152831180751455699556598978, 5.47592264342608419892298603881, 5.95390210477292352580192271917
