| L(s) = 1 | − 2·3-s − 4·5-s + 2·7-s + 2·11-s + 4·13-s + 8·15-s − 4·17-s − 6·19-s − 4·21-s − 14·23-s + 10·25-s + 6·27-s + 4·29-s − 10·31-s − 4·33-s − 8·35-s − 16·37-s − 8·39-s − 14·43-s + 6·47-s − 16·49-s + 8·51-s + 8·53-s − 8·55-s + 12·57-s + 12·59-s + 16·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 0.755·7-s + 0.603·11-s + 1.10·13-s + 2.06·15-s − 0.970·17-s − 1.37·19-s − 0.872·21-s − 2.91·23-s + 2·25-s + 1.15·27-s + 0.742·29-s − 1.79·31-s − 0.696·33-s − 1.35·35-s − 2.63·37-s − 1.28·39-s − 2.13·43-s + 0.875·47-s − 2.28·49-s + 1.12·51-s + 1.09·53-s − 1.07·55-s + 1.58·57-s + 1.56·59-s + 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T + 4 T^{2} + 2 T^{3} + 2 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.c_e_c_c |
| 7 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 26 T^{3} + 178 T^{4} - 26 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ac_u_aba_gw |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 34 T^{3} + 218 T^{4} - 34 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_u_abi_ik |
| 13 | $C_2 \wr S_4$ | \( 1 - 4 T + 28 T^{2} - 108 T^{3} + 550 T^{4} - 108 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_bc_aee_ve |
| 17 | $C_2 \wr S_4$ | \( 1 + 4 T + 28 T^{2} + 88 T^{3} + 674 T^{4} + 88 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_bc_dk_zy |
| 19 | $C_2 \wr S_4$ | \( 1 + 6 T + 48 T^{2} + 182 T^{3} + 1138 T^{4} + 182 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.g_bw_ha_bru |
| 23 | $C_2 \wr S_4$ | \( 1 + 14 T + 120 T^{2} + 710 T^{3} + 3690 T^{4} + 710 p T^{5} + 120 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.o_eq_bbi_fly |
| 31 | $C_2 \wr S_4$ | \( 1 + 10 T + 120 T^{2} + 834 T^{3} + 5618 T^{4} + 834 p T^{5} + 120 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.k_eq_bgc_iic |
| 37 | $C_2 \wr S_4$ | \( 1 + 16 T + 196 T^{2} + 1572 T^{3} + 11170 T^{4} + 1572 p T^{5} + 196 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.q_ho_cim_qnq |
| 41 | $C_2 \wr S_4$ | \( 1 + 92 T^{2} + 112 T^{3} + 4662 T^{4} + 112 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_do_ei_gxi |
| 43 | $C_2 \wr S_4$ | \( 1 + 14 T + 236 T^{2} + 1918 T^{3} + 16658 T^{4} + 1918 p T^{5} + 236 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.o_jc_cvu_yqs |
| 47 | $C_2 \wr S_4$ | \( 1 - 6 T + 148 T^{2} - 566 T^{3} + 9122 T^{4} - 566 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ag_fs_avu_nmw |
| 53 | $C_2 \wr S_4$ | \( 1 - 8 T + 188 T^{2} - 1128 T^{3} + 14662 T^{4} - 1128 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ai_hg_abrk_vry |
| 59 | $C_2 \wr S_4$ | \( 1 - 12 T + 220 T^{2} - 1756 T^{3} + 18278 T^{4} - 1756 p T^{5} + 220 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.am_im_acpo_bbba |
| 61 | $C_2 \wr S_4$ | \( 1 - 16 T + 228 T^{2} - 1696 T^{3} + 15670 T^{4} - 1696 p T^{5} + 228 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aq_iu_acng_xes |
| 67 | $C_2 \wr S_4$ | \( 1 + 6 T + 64 T^{2} + 790 T^{3} + 8122 T^{4} + 790 p T^{5} + 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.g_cm_bek_mak |
| 71 | $C_2 \wr S_4$ | \( 1 - 4 T + 252 T^{2} - 836 T^{3} + 25750 T^{4} - 836 p T^{5} + 252 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ae_js_abge_bmck |
| 73 | $C_2 \wr S_4$ | \( 1 - 12 T + 192 T^{2} - 1144 T^{3} + 14554 T^{4} - 1144 p T^{5} + 192 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.am_hk_absa_vnu |
| 79 | $C_2 \wr S_4$ | \( 1 + 2 T + 180 T^{2} + 1170 T^{3} + 14954 T^{4} + 1170 p T^{5} + 180 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.c_gy_bta_wde |
| 83 | $C_2 \wr S_4$ | \( 1 + 22 T + 428 T^{2} + 5222 T^{3} + 55746 T^{4} + 5222 p T^{5} + 428 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.w_qm_hsw_demc |
| 89 | $C_2 \wr S_4$ | \( 1 - 4 T + 292 T^{2} - 988 T^{3} + 36118 T^{4} - 988 p T^{5} + 292 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ae_lg_abma_cble |
| 97 | $C_2 \wr S_4$ | \( 1 + 8 T + 160 T^{2} - 660 T^{3} + 3130 T^{4} - 660 p T^{5} + 160 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.i_ge_azk_eqk |
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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
−6.32886982281925727464834264972, −6.11382954981828942982688387454, −5.84352406839627860249171529714, −5.74270698859449868267301574406, −5.43030842351825064909182917484, −5.26963263120872884135803433901, −5.01706257839738349293114420650, −4.88049098752780272447941074997, −4.71795051335093248368932457148, −4.43606771940509167137488995285, −4.43042707441071340243379712334, −4.01651121550546375078762179791, −3.73872792067146734546081135474, −3.65505314487590877519556392531, −3.62504171790844586086942909643, −3.48957385195856211239367362011, −3.33630579973758054167248160065, −2.52217401091633084382010973675, −2.45456953219867670671976048995, −2.35979034570074809410293130743, −2.11534299745231512805682488857, −1.74443086165400559916698661029, −1.35066792357052187887798084806, −1.14147876908417224576733272584, −1.12876278536777126559296543491, 0, 0, 0, 0,
1.12876278536777126559296543491, 1.14147876908417224576733272584, 1.35066792357052187887798084806, 1.74443086165400559916698661029, 2.11534299745231512805682488857, 2.35979034570074809410293130743, 2.45456953219867670671976048995, 2.52217401091633084382010973675, 3.33630579973758054167248160065, 3.48957385195856211239367362011, 3.62504171790844586086942909643, 3.65505314487590877519556392531, 3.73872792067146734546081135474, 4.01651121550546375078762179791, 4.43042707441071340243379712334, 4.43606771940509167137488995285, 4.71795051335093248368932457148, 4.88049098752780272447941074997, 5.01706257839738349293114420650, 5.26963263120872884135803433901, 5.43030842351825064909182917484, 5.74270698859449868267301574406, 5.84352406839627860249171529714, 6.11382954981828942982688387454, 6.32886982281925727464834264972
