| L(s) = 1 | − 3-s − 2·5-s − 2·7-s − 2·13-s + 2·15-s + 4·17-s − 6·19-s + 2·21-s + 5·25-s − 8·29-s + 8·31-s + 4·35-s − 10·37-s + 2·39-s + 8·41-s + 8·43-s + 8·47-s + 7·49-s − 4·51-s + 2·53-s + 6·57-s − 12·59-s + 10·61-s + 4·65-s + 48·67-s − 8·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s − 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s + 25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 0.320·39-s + 1.24·41-s + 1.21·43-s + 1.16·47-s + 49-s − 0.560·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 5.86·67-s − 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7060669608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7060669608\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) | |
| 11 | | \( 1 \) | |
| good | 5 | $C_4\times C_2$ | \( 1 + 2 T - T^{2} - 12 T^{3} - 19 T^{4} - 12 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.c_ab_am_at |
| 7 | $C_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_ad_au_at |
| 13 | $C_4\times C_2$ | \( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 44 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.c_aj_abs_bd |
| 17 | $C_4\times C_2$ | \( 1 - 4 T - T^{2} + 72 T^{3} - 271 T^{4} + 72 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_ab_cu_akl |
| 19 | $C_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 12 T^{3} - 395 T^{4} - 12 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.g_r_am_apf |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.23.a_do_a_esc |
| 29 | $C_4\times C_2$ | \( 1 + 8 T + 35 T^{2} + 48 T^{3} - 631 T^{4} + 48 p T^{5} + 35 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.i_bj_bw_ayh |
| 31 | $C_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 16 p T^{5} + 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ai_bh_aq_abil |
| 37 | $C_4\times C_2$ | \( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} + 260 p T^{5} + 63 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.k_cl_ka_kj |
| 41 | $C_4\times C_2$ | \( 1 - 8 T + 23 T^{2} + 144 T^{3} - 2095 T^{4} + 144 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ai_x_fo_adcp |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.43.ai_ho_aboy_tms |
| 47 | $C_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 240 T^{3} - 2719 T^{4} + 240 p T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ai_r_jg_aeap |
| 53 | $C_4\times C_2$ | \( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 204 p T^{5} - 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ac_abx_hw_dgf |
| 59 | $C_4\times C_2$ | \( 1 + 12 T + 85 T^{2} + 312 T^{3} - 1271 T^{4} + 312 p T^{5} + 85 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.m_dh_ma_abwx |
| 61 | $C_4\times C_2$ | \( 1 - 10 T + 39 T^{2} + 220 T^{3} - 4579 T^{4} + 220 p T^{5} + 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ak_bn_im_agud |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) | 4.67.abw_bro_aymy_jhuk |
| 71 | $C_4\times C_2$ | \( 1 + 8 T - 7 T^{2} - 624 T^{3} - 4495 T^{4} - 624 p T^{5} - 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.i_ah_aya_agqx |
| 73 | $C_4\times C_2$ | \( 1 - 6 T - 37 T^{2} + 660 T^{3} - 1259 T^{4} + 660 p T^{5} - 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ag_abl_zk_abwl |
| 79 | $C_4\times C_2$ | \( 1 + 2 T - 75 T^{2} - 308 T^{3} + 5309 T^{4} - 308 p T^{5} - 75 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.c_acx_alw_hwf |
| 83 | $C_4\times C_2$ | \( 1 - 16 T + 173 T^{2} - 1440 T^{3} + 8681 T^{4} - 1440 p T^{5} + 173 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.aq_gr_acdk_mvx |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) | 4.89.ce_cgy_bmjg_quuo |
| 97 | $C_4\times C_2$ | \( 1 - 2 T - 93 T^{2} + 380 T^{3} + 8261 T^{4} + 380 p T^{5} - 93 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ac_adp_oq_mft |
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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
−6.82219415147594354514704203396, −6.58332392981433637441459402926, −6.34163580845518798933321926450, −6.05344089390208926915169429378, −6.04146578886830814708804025182, −5.75300427402332193600866654536, −5.38719911685796973642552285071, −5.25547092700228855944949242934, −5.15504140721396626878186882579, −4.79228532792667038523104901934, −4.67331480641919905769181816668, −4.04223008806573519447715618035, −3.98842386049649381647648095656, −3.91019655259756609795298375505, −3.90966043877810159404659369320, −3.30504530269328479623520295577, −3.13494144292016723035188783477, −2.76472406914376579347586199674, −2.46554484556488747519190646673, −2.27921121142361240423258248626, −2.12134652514468718097713700167, −1.38603074627643692759832738212, −1.12658117661863178547711105650, −0.70585459925267688342960609418, −0.23937346365887367998151966245,
0.23937346365887367998151966245, 0.70585459925267688342960609418, 1.12658117661863178547711105650, 1.38603074627643692759832738212, 2.12134652514468718097713700167, 2.27921121142361240423258248626, 2.46554484556488747519190646673, 2.76472406914376579347586199674, 3.13494144292016723035188783477, 3.30504530269328479623520295577, 3.90966043877810159404659369320, 3.91019655259756609795298375505, 3.98842386049649381647648095656, 4.04223008806573519447715618035, 4.67331480641919905769181816668, 4.79228532792667038523104901934, 5.15504140721396626878186882579, 5.25547092700228855944949242934, 5.38719911685796973642552285071, 5.75300427402332193600866654536, 6.04146578886830814708804025182, 6.05344089390208926915169429378, 6.34163580845518798933321926450, 6.58332392981433637441459402926, 6.82219415147594354514704203396
