| L(s) = 1 | + 2·2-s − 4·3-s + 2·4-s + 4·5-s − 8·6-s + 4·7-s + 10·9-s + 8·10-s + 6·11-s − 8·12-s + 2·13-s + 8·14-s − 16·15-s − 3·16-s − 10·17-s + 20·18-s + 16·19-s + 8·20-s − 16·21-s + 12·22-s + 6·23-s + 10·25-s + 4·26-s − 20·27-s + 8·28-s − 32·30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 4-s + 1.78·5-s − 3.26·6-s + 1.51·7-s + 10/3·9-s + 2.52·10-s + 1.80·11-s − 2.30·12-s + 0.554·13-s + 2.13·14-s − 4.13·15-s − 3/4·16-s − 2.42·17-s + 4.71·18-s + 3.67·19-s + 1.78·20-s − 3.49·21-s + 2.55·22-s + 1.25·23-s + 2·25-s + 0.784·26-s − 3.84·27-s + 1.51·28-s − 5.84·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.870088714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.870088714\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p T + p T^{2} - T^{4} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_c_a_ab |
| 7 | $C_2 \wr S_4$ | \( 1 - 4 T + 18 T^{2} - 38 T^{3} + 122 T^{4} - 38 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ae_s_abm_es |
| 11 | $C_2 \wr S_4$ | \( 1 - 6 T + 32 T^{2} - 142 T^{3} + 494 T^{4} - 142 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ag_bg_afm_ta |
| 13 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 T^{2} + 16 T^{3} - 282 T^{4} + 16 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ac_c_q_akw |
| 17 | $C_2 \wr S_4$ | \( 1 + 10 T + 88 T^{2} + 458 T^{3} + 134 p T^{4} + 458 p T^{5} + 88 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.k_dk_rq_djq |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.19.aq_gq_absy_izm |
| 23 | $C_2 \wr S_4$ | \( 1 - 6 T + 8 T^{2} - 122 T^{3} + 1358 T^{4} - 122 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ag_i_aes_cag |
| 29 | $C_2 \wr S_4$ | \( 1 + 104 T^{2} + 14 T^{3} + 4346 T^{4} + 14 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ea_o_gle |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 122 T^{2} - 714 T^{3} + 6678 T^{4} - 714 p T^{5} + 122 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_es_abbm_jww |
| 41 | $C_2 \wr S_4$ | \( 1 + 6 T + 100 T^{2} + 258 T^{3} + 4134 T^{4} + 258 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.g_dw_jy_gda |
| 43 | $C_2 \wr S_4$ | \( 1 - 2 T + 80 T^{2} - 26 T^{3} + 4110 T^{4} - 26 p T^{5} + 80 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ac_dc_aba_gcc |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 80 T^{2} - 306 T^{3} + 2366 T^{4} - 306 p T^{5} + 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.c_dc_alu_dna |
| 53 | $C_2 \wr S_4$ | \( 1 + 6 T + 204 T^{2} + 934 T^{3} + 15998 T^{4} + 934 p T^{5} + 204 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.g_hw_bjy_xri |
| 59 | $C_2 \wr S_4$ | \( 1 - 4 T + 128 T^{2} - 78 T^{3} + 7334 T^{4} - 78 p T^{5} + 128 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ae_ey_ada_kwc |
| 61 | $C_2 \wr S_4$ | \( 1 + 76 T^{2} + 608 T^{3} + 2678 T^{4} + 608 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_cy_xk_dza |
| 67 | $C_2 \wr S_4$ | \( 1 + 2 T + 202 T^{2} + 200 T^{3} + 18274 T^{4} + 200 p T^{5} + 202 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.c_hu_hs_bbaw |
| 71 | $C_2 \wr S_4$ | \( 1 - 22 T + 372 T^{2} - 4152 T^{3} + 40934 T^{4} - 4152 p T^{5} + 372 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aw_oi_agds_ciok |
| 73 | $C_2 \wr S_4$ | \( 1 + 32 T + 634 T^{2} + 8298 T^{3} + 82694 T^{4} + 8298 p T^{5} + 634 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.bg_yk_mhe_esio |
| 79 | $C_2 \wr S_4$ | \( 1 - 24 T + 248 T^{2} - 1044 T^{3} + 2638 T^{4} - 1044 p T^{5} + 248 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ay_jo_aboe_dxm |
| 83 | $C_2 \wr S_4$ | \( 1 + 6 T + 288 T^{2} + 1154 T^{3} + 33566 T^{4} + 1154 p T^{5} + 288 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.g_lc_bsk_bxra |
| 89 | $C_2 \wr S_4$ | \( 1 + 14 T + 296 T^{2} + 3256 T^{3} + 36754 T^{4} + 3256 p T^{5} + 296 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.o_lk_evg_ccjq |
| 97 | $C_2 \wr S_4$ | \( 1 + 14 T + 436 T^{2} + 42 p T^{3} + 65702 T^{4} + 42 p^{2} T^{5} + 436 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.o_qu_gas_dtfa |
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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
−7.77999332001281975653138138238, −7.57659076213834897420278428115, −7.03128626279949067866236527715, −7.02657973620908077679462760817, −7.01460396633596503441791904819, −6.62867761598136711058410123255, −6.38966625221687898553740334682, −5.98703412879406635404465388987, −5.96670521489494042438475920445, −5.78269881570778371135926794523, −5.38351116458437659306573159768, −5.17880064341867373810871121357, −5.16369197364004096883377848812, −4.61274457673710009836924025056, −4.59953313709569860037675744980, −4.25471806108173832146126025913, −4.24486273524757213644383420141, −3.26713990809753323267736899422, −3.24712780492550970222985314375, −3.18990048675742561578658892536, −2.17887318093600887399742421013, −1.99734687363675633370292512642, −1.63283863427970266377203935420, −1.07216669848970112173245795198, −0.995451031500777909101241142893,
0.995451031500777909101241142893, 1.07216669848970112173245795198, 1.63283863427970266377203935420, 1.99734687363675633370292512642, 2.17887318093600887399742421013, 3.18990048675742561578658892536, 3.24712780492550970222985314375, 3.26713990809753323267736899422, 4.24486273524757213644383420141, 4.25471806108173832146126025913, 4.59953313709569860037675744980, 4.61274457673710009836924025056, 5.16369197364004096883377848812, 5.17880064341867373810871121357, 5.38351116458437659306573159768, 5.78269881570778371135926794523, 5.96670521489494042438475920445, 5.98703412879406635404465388987, 6.38966625221687898553740334682, 6.62867761598136711058410123255, 7.01460396633596503441791904819, 7.02657973620908077679462760817, 7.03128626279949067866236527715, 7.57659076213834897420278428115, 7.77999332001281975653138138238
