| L(s) = 1 | − 4·2-s + 4·3-s + 10·4-s − 16·6-s + 7·7-s − 20·8-s + 10·9-s + 9·11-s + 40·12-s + 4·13-s − 28·14-s + 35·16-s + 6·17-s − 40·18-s + 4·19-s + 28·21-s − 36·22-s + 24·23-s − 80·24-s − 5·25-s − 16·26-s + 20·27-s + 70·28-s + 15·29-s − 56·32-s + 36·33-s − 24·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 6.53·6-s + 2.64·7-s − 7.07·8-s + 10/3·9-s + 2.71·11-s + 11.5·12-s + 1.10·13-s − 7.48·14-s + 35/4·16-s + 1.45·17-s − 9.42·18-s + 0.917·19-s + 6.11·21-s − 7.67·22-s + 5.00·23-s − 16.3·24-s − 25-s − 3.13·26-s + 3.84·27-s + 13.2·28-s + 2.78·29-s − 9.89·32-s + 6.26·33-s − 4.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.94511765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.94511765\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 31 | | \( 1 \) | |
| good | 5 | $C_2^2:C_4$ | \( 1 + p T^{2} + 9 p T^{4} + p^{3} T^{6} + p^{4} T^{8} \) | 4.5.a_f_a_bt |
| 7 | $((C_8 : C_2):C_2):C_2$ | \( 1 - p T + 6 p T^{2} - 155 T^{3} + 491 T^{4} - 155 p T^{5} + 6 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.7.ah_bq_afz_sx |
| 11 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 9 T + 50 T^{2} - 171 T^{3} + 579 T^{4} - 171 p T^{5} + 50 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.aj_by_agp_wh |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ae_bw_afk_bjq |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 6 T + 44 T^{2} - 162 T^{3} + 774 T^{4} - 162 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_bs_agg_bdu |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ae_cu_aie_dac |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.23.ay_lw_adsy_vic |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 15 T + 176 T^{2} - 1395 T^{3} + 8571 T^{4} - 1395 p T^{5} + 176 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ap_gu_acbr_mrr |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 132 T^{2} + 880 T^{3} + 7046 T^{4} + 880 p T^{5} + 132 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.i_fc_bhw_kla |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 12 T + 68 T^{2} + 684 T^{3} + 6390 T^{4} + 684 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.m_cq_bai_jlu |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 4 T + 108 T^{2} - 140 T^{3} + 5126 T^{4} - 140 p T^{5} + 108 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_ee_afk_hpe |
| 47 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 68 T^{2} + 360 T^{3} + 2694 T^{4} + 360 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_cq_nw_dzq |
| 53 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 3 T + 56 T^{2} + 99 T^{3} + 3819 T^{4} + 99 p T^{5} + 56 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.d_ce_dv_fqx |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 3 T + 170 T^{2} - 423 T^{3} + 13899 T^{4} - 423 p T^{5} + 170 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ad_go_aqh_uop |
| 61 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 14 T + 240 T^{2} + 2026 T^{3} + 20414 T^{4} + 2026 p T^{5} + 240 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.o_jg_czy_befe |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 2 T + 192 T^{2} + 250 T^{3} + 17726 T^{4} + 250 p T^{5} + 192 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.c_hk_jq_bafu |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 30 T + 524 T^{2} - 90 p T^{3} + 60726 T^{4} - 90 p^{2} T^{5} + 524 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.abe_ue_ajlu_dlvq |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 26 T + 483 T^{2} + 6040 T^{3} + 60041 T^{4} + 6040 p T^{5} + 483 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ba_sp_iyi_dkvh |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 10 T + 81 T^{2} + 1030 T^{3} - 10909 T^{4} + 1030 p T^{5} + 81 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ak_dd_bnq_aqdp |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 6 T + 248 T^{2} - 720 T^{3} + 25761 T^{4} - 720 p T^{5} + 248 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ag_jo_abbs_bmcv |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 6 T + 92 T^{2} - 702 T^{3} + 390 T^{4} - 702 p T^{5} + 92 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.g_do_abba_pa |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 4 T + 114 T^{2} - 1568 T^{3} + 11099 T^{4} - 1568 p T^{5} + 114 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ae_ek_acii_qkx |
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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
−5.77743296263060646974184465592, −5.51917876292462419677093877225, −5.28340526336684800818441701322, −5.10263988984445660229811807911, −4.95565955695836819829118119542, −4.68782212714291078939146231540, −4.52996196342124103087428697889, −4.42210656379814892846364066968, −4.16803613596326172109001031330, −3.52263725526703522833534135860, −3.50198636299780347340402843130, −3.44736615162674247720729572878, −3.35158630866265349092081912876, −2.98328499416059921488091468135, −2.96761961026543120355638581400, −2.64196463751372250402682319323, −2.31480036162628884834413642072, −1.86042290744864189012978427702, −1.76596800670575253154555624697, −1.66311345430413111568100830430, −1.50035759980673719094557129431, −1.14761860973336991774983806289, −0.997012737527422931360948001132, −0.937641238084733013172301207714, −0.70306690067246566161270266489,
0.70306690067246566161270266489, 0.937641238084733013172301207714, 0.997012737527422931360948001132, 1.14761860973336991774983806289, 1.50035759980673719094557129431, 1.66311345430413111568100830430, 1.76596800670575253154555624697, 1.86042290744864189012978427702, 2.31480036162628884834413642072, 2.64196463751372250402682319323, 2.96761961026543120355638581400, 2.98328499416059921488091468135, 3.35158630866265349092081912876, 3.44736615162674247720729572878, 3.50198636299780347340402843130, 3.52263725526703522833534135860, 4.16803613596326172109001031330, 4.42210656379814892846364066968, 4.52996196342124103087428697889, 4.68782212714291078939146231540, 4.95565955695836819829118119542, 5.10263988984445660229811807911, 5.28340526336684800818441701322, 5.51917876292462419677093877225, 5.77743296263060646974184465592
