| L(s) = 1 | + 3-s + 9·7-s + 5·11-s − 4·13-s − 4·17-s + 9·21-s + 11·23-s − 2·27-s + 10·29-s + 9·31-s + 5·33-s − 15·37-s − 4·39-s + 17·41-s − 12·43-s + 14·47-s + 35·49-s − 4·51-s + 6·53-s + 18·59-s + 11·61-s + 67-s + 11·69-s + 26·71-s − 9·73-s + 45·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.40·7-s + 1.50·11-s − 1.10·13-s − 0.970·17-s + 1.96·21-s + 2.29·23-s − 0.384·27-s + 1.85·29-s + 1.61·31-s + 0.870·33-s − 2.46·37-s − 0.640·39-s + 2.65·41-s − 1.82·43-s + 2.04·47-s + 5·49-s − 0.560·51-s + 0.824·53-s + 2.34·59-s + 1.40·61-s + 0.122·67-s + 1.32·69-s + 3.08·71-s − 1.05·73-s + 5.12·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.29048809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.29048809\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + T^{2} + T^{3} + 8 T^{4} + p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ab_b_b_i |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 46 T^{2} - 172 T^{3} + 515 T^{4} - 172 p T^{5} + 46 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.aj_bu_agq_tv |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 42 T^{2} - 140 T^{3} + 683 T^{4} - 140 p T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.af_bq_afk_bah |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 35 T^{2} + 154 T^{3} + 591 T^{4} + 154 p T^{5} + 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.e_bj_fy_wt |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 47 T^{2} + 104 T^{3} + 940 T^{4} + 104 p T^{5} + 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_bv_ea_bke |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T^{2} + 110 T^{3} + 191 T^{4} + 110 p T^{5} + p^{3} T^{6} + p^{4} T^{8} \) | 4.19.a_t_eg_hj |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 95 T^{2} - 561 T^{3} + 2996 T^{4} - 561 p T^{5} + 95 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.al_dr_avp_elg |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 123 T^{2} - 860 T^{3} + 5448 T^{4} - 860 p T^{5} + 123 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ak_et_abhc_ibo |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 73 T^{2} - 127 T^{3} + 824 T^{4} - 127 p T^{5} + 73 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.aj_cv_aex_bfs |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 15 T + 191 T^{2} + 1535 T^{3} + 10992 T^{4} + 1535 p T^{5} + 191 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.p_hj_chb_qgu |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 17 T + 166 T^{2} - 1432 T^{3} + 10389 T^{4} - 1432 p T^{5} + 166 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ar_gk_acdc_pjp |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 199 T^{2} + 1544 T^{3} + 13400 T^{4} + 1544 p T^{5} + 199 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.m_hr_chk_tvk |
| 47 | $D_{4}$ | \( ( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ao_hr_acns_vrl |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 113 T^{2} - 230 T^{3} + 5651 T^{4} - 230 p T^{5} + 113 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ag_ej_aiw_ijj |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 18 T + 333 T^{2} - 3330 T^{3} + 32271 T^{4} - 3330 p T^{5} + 333 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.as_mv_aeyc_bvtf |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 273 T^{2} - 1993 T^{3} + 25764 T^{4} - 1993 p T^{5} + 273 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.al_kn_acyr_bmcy |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 71 T^{2} - 183 T^{3} + 9060 T^{4} - 183 p T^{5} + 71 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ab_ct_ahb_nkm |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 26 T + 423 T^{2} - 5148 T^{3} + 49964 T^{4} - 5148 p T^{5} + 423 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aba_qh_ahqa_cvxs |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 185 T^{2} + 1489 T^{3} + 18876 T^{4} + 1489 p T^{5} + 185 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.j_hd_cfh_bbya |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 227 T^{2} + 1456 T^{3} + 25804 T^{4} + 1456 p T^{5} + 227 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.i_it_cea_bmem |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 144 T^{2} + 364 T^{3} + 4430 T^{4} + 364 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.m_fo_oa_gok |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 159 T^{2} + 245 T^{3} + 11376 T^{4} + 245 p T^{5} + 159 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.af_gd_jl_qvo |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 29 T + 539 T^{2} + 7133 T^{3} + 76944 T^{4} + 7133 p T^{5} + 539 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bd_ut_koj_ejvk |
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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
−7.16769043865944030227925991733, −6.90041198406868709937139498054, −6.87816527923958316140186228097, −6.63483316616915228765245408933, −6.46029079474752431194078208863, −5.84234553388715514424337504486, −5.54107741129948994577262154335, −5.48618407129202226683476851934, −5.46021367744418967326168074788, −4.81687255541543521495691303559, −4.75743668499902454755646562203, −4.60798852650251484423786111403, −4.49947537809275349032511636308, −4.31096963388915290782774110453, −3.73441072697800185501397650499, −3.67162476586240725046879456243, −3.37704031760876016344534896195, −2.74227871924204691855909886220, −2.58275474489678653850896645542, −2.37491062779214720749475330098, −2.15096214265904922896510184099, −1.64344288522645581589881758346, −1.41032635844019483646908503044, −0.900366202041063146447159861662, −0.883275467342004914901908137896,
0.883275467342004914901908137896, 0.900366202041063146447159861662, 1.41032635844019483646908503044, 1.64344288522645581589881758346, 2.15096214265904922896510184099, 2.37491062779214720749475330098, 2.58275474489678653850896645542, 2.74227871924204691855909886220, 3.37704031760876016344534896195, 3.67162476586240725046879456243, 3.73441072697800185501397650499, 4.31096963388915290782774110453, 4.49947537809275349032511636308, 4.60798852650251484423786111403, 4.75743668499902454755646562203, 4.81687255541543521495691303559, 5.46021367744418967326168074788, 5.48618407129202226683476851934, 5.54107741129948994577262154335, 5.84234553388715514424337504486, 6.46029079474752431194078208863, 6.63483316616915228765245408933, 6.87816527923958316140186228097, 6.90041198406868709937139498054, 7.16769043865944030227925991733
