| L(s) = 1 | + 3·2-s + 4·4-s − 4·5-s − 11·7-s + 4·8-s − 12·10-s − 7·13-s − 33·14-s + 2·16-s + 3·17-s − 12·19-s − 16·20-s + 9·23-s + 10·25-s − 21·26-s − 44·28-s − 8·29-s + 3·31-s − 5·32-s + 9·34-s + 44·35-s − 3·37-s − 36·38-s − 16·40-s − 7·41-s − 21·43-s + 27·46-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s − 1.78·5-s − 4.15·7-s + 1.41·8-s − 3.79·10-s − 1.94·13-s − 8.81·14-s + 1/2·16-s + 0.727·17-s − 2.75·19-s − 3.57·20-s + 1.87·23-s + 2·25-s − 4.11·26-s − 8.31·28-s − 1.48·29-s + 0.538·31-s − 0.883·32-s + 1.54·34-s + 7.43·35-s − 0.493·37-s − 5.83·38-s − 2.52·40-s − 1.09·41-s − 3.20·43-s + 3.98·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $(((C_4 \times C_2): C_2):C_2):C_2$ | \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 11 T^{4} - 7 p T^{5} + 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ad_f_ah_l |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 11 T + 67 T^{2} + 276 T^{3} + 845 T^{4} + 276 p T^{5} + 67 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.l_cp_kq_bgn |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 59 T^{2} + 264 T^{3} + 1185 T^{4} + 264 p T^{5} + 59 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.h_ch_ke_btp |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 60 T^{2} - 127 T^{3} + 1451 T^{4} - 127 p T^{5} + 60 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ad_ci_aex_cdv |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 122 T^{2} + 749 T^{3} + 3939 T^{4} + 749 p T^{5} + 122 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.m_es_bcv_fvn |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 38 T^{2} + 85 T^{3} - 979 T^{4} + 85 p T^{5} + 38 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.aj_bm_dh_ablr |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 112 T^{2} + 601 T^{3} + 4759 T^{4} + 601 p T^{5} + 112 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.i_ei_xd_hbb |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 3 p T^{2} - 276 T^{3} + 3945 T^{4} - 276 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ad_dp_akq_fvt |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 92 T^{2} + 305 T^{3} + 4221 T^{4} + 305 p T^{5} + 92 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.d_do_lt_ggj |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 118 T^{2} + 479 T^{3} + 5815 T^{4} + 479 p T^{5} + 118 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.h_eo_sl_ipr |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 21 T + 293 T^{2} + 2900 T^{3} + 21441 T^{4} + 2900 p T^{5} + 293 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.v_lh_eho_bfsr |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 137 T^{2} - 290 T^{3} + 8531 T^{4} - 290 p T^{5} + 137 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ad_fh_ale_mqd |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 245 T^{2} - 1776 T^{3} + 20351 T^{4} - 1776 p T^{5} + 245 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.al_jl_acqi_bect |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 127 T^{2} - 44 T^{3} + 4999 T^{4} - 44 p T^{5} + 127 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ah_ex_abs_hkh |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 203 T^{2} + 642 T^{3} + 17379 T^{4} + 642 p T^{5} + 203 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.e_hv_ys_zsl |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.67.b_he_abl_xll |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 186 T^{2} - 925 T^{3} + 8531 T^{4} - 925 p T^{5} + 186 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ap_he_abjp_mqd |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 218 T^{2} - 1375 T^{3} + 20781 T^{4} - 1375 p T^{5} + 218 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aj_ik_acax_beth |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 220 T^{2} + 487 T^{3} + 20123 T^{4} + 487 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.g_im_st_bdtz |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 375 T^{2} - 3710 T^{3} + 48443 T^{4} - 3710 p T^{5} + 375 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ap_ol_afms_ctrf |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 206 T^{2} + 400 T^{3} + 21551 T^{4} + 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_hy_pk_bfwx |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 332 T^{2} - 1836 T^{3} + 45565 T^{4} - 1836 p T^{5} + 332 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ag_mu_acsq_cpkn |
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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
−6.31266864358770585329950047141, −5.89413475414355669711994060037, −5.63873428978478738912814950688, −5.52282745522235805010257332514, −5.30635487483649972249678156655, −4.90641821070624806701040168734, −4.87691193734607554542190152896, −4.83198135104809799322650187902, −4.82918236667583376469132844578, −4.27506024384880215156206744113, −3.95445317624459373917671416524, −3.94117188056168071599633884566, −3.89524910630540932146716706159, −3.65359006681426002269949702867, −3.43319816094574749187161383917, −3.28596499633964944538692789323, −3.16807974868779073627625513177, −2.80626952446152239715761460035, −2.66573352250736282721809380087, −2.61683345888984826272506584252, −2.21463854583247879323979786231, −1.92672903141222081622690751279, −1.76104546058491359891943200085, −0.985874326017581328863974864529, −0.908178122001341752123187369260, 0, 0, 0, 0,
0.908178122001341752123187369260, 0.985874326017581328863974864529, 1.76104546058491359891943200085, 1.92672903141222081622690751279, 2.21463854583247879323979786231, 2.61683345888984826272506584252, 2.66573352250736282721809380087, 2.80626952446152239715761460035, 3.16807974868779073627625513177, 3.28596499633964944538692789323, 3.43319816094574749187161383917, 3.65359006681426002269949702867, 3.89524910630540932146716706159, 3.94117188056168071599633884566, 3.95445317624459373917671416524, 4.27506024384880215156206744113, 4.82918236667583376469132844578, 4.83198135104809799322650187902, 4.87691193734607554542190152896, 4.90641821070624806701040168734, 5.30635487483649972249678156655, 5.52282745522235805010257332514, 5.63873428978478738912814950688, 5.89413475414355669711994060037, 6.31266864358770585329950047141
