| L(s) = 1 | − 5·2-s − 3·3-s + 15·4-s + 4·5-s + 15·6-s − 5·7-s − 35·8-s − 3·9-s − 20·10-s − 45·12-s − 3·13-s + 25·14-s − 12·15-s + 70·16-s − 17-s + 15·18-s + 4·19-s + 60·20-s + 15·21-s + 105·24-s + 15·26-s + 24·27-s − 75·28-s − 10·29-s + 60·30-s − 15·31-s − 126·32-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 1.73·3-s + 15/2·4-s + 1.78·5-s + 6.12·6-s − 1.88·7-s − 12.3·8-s − 9-s − 6.32·10-s − 12.9·12-s − 0.832·13-s + 6.68·14-s − 3.09·15-s + 35/2·16-s − 0.242·17-s + 3.53·18-s + 0.917·19-s + 13.4·20-s + 3.27·21-s + 21.4·24-s + 2.94·26-s + 4.61·27-s − 14.1·28-s − 1.85·29-s + 10.9·30-s − 2.69·31-s − 22.2·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + p T + 4 p T^{2} + 7 p T^{3} + 53 T^{4} + 73 T^{5} + 53 p T^{6} + 7 p^{3} T^{7} + 4 p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} \) | 5.3.d_m_v_cb_cv |
| 5 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 4 T + 16 T^{2} - 42 T^{3} + 27 p T^{4} - 309 T^{5} + 27 p^{2} T^{6} - 42 p^{2} T^{7} + 16 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.5.ae_q_abq_ff_alx |
| 11 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 3 p T^{2} - p T^{3} + 48 p T^{4} - 21 p T^{5} + 48 p^{2} T^{6} - p^{3} T^{7} + 3 p^{4} T^{8} + p^{5} T^{10} \) | 5.11.a_bh_al_ui_aix |
| 13 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 3 T + 62 T^{2} + 141 T^{3} + 1563 T^{4} + 2653 T^{5} + 1563 p T^{6} + 141 p^{2} T^{7} + 62 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.d_ck_fl_cid_dyb |
| 17 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + T + 59 T^{2} + 43 T^{3} + 1655 T^{4} + 951 T^{5} + 1655 p T^{6} + 43 p^{2} T^{7} + 59 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.17.b_ch_br_clr_bkp |
| 19 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 4 T + 86 T^{2} - 299 T^{3} + 3106 T^{4} - 8475 T^{5} + 3106 p T^{6} - 299 p^{2} T^{7} + 86 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.ae_di_aln_epm_amnz |
| 29 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 10 T + 152 T^{2} + 987 T^{3} + 8538 T^{4} + 40073 T^{5} + 8538 p T^{6} + 987 p^{2} T^{7} + 152 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.k_fw_blz_mqk_chhh |
| 31 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 15 T + 234 T^{2} + 2020 T^{3} + 17010 T^{4} + 96301 T^{5} + 17010 p T^{6} + 2020 p^{2} T^{7} + 234 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.p_ja_czs_zeg_fmlx |
| 37 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 3 T + 105 T^{2} + 121 T^{3} + 4767 T^{4} + 829 T^{5} + 4767 p T^{6} + 121 p^{2} T^{7} + 105 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.d_eb_er_hbj_bfx |
| 41 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 6 T + 61 T^{2} - 159 T^{3} - 930 T^{4} - 24631 T^{5} - 930 p T^{6} - 159 p^{2} T^{7} + 61 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.g_cj_agd_abju_abklj |
| 43 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 16 T + 291 T^{2} - 2817 T^{3} + 28112 T^{4} - 183161 T^{5} + 28112 p T^{6} - 2817 p^{2} T^{7} + 291 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.aq_lf_aeej_bppg_akkyr |
| 47 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 19 T + 320 T^{2} - 3279 T^{3} + 31293 T^{4} - 219641 T^{5} + 31293 p T^{6} - 3279 p^{2} T^{7} + 320 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.at_mi_aewd_buhp_ammxt |
| 53 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 14 T + 196 T^{2} - 1545 T^{3} + 14260 T^{4} - 92137 T^{5} + 14260 p T^{6} - 1545 p^{2} T^{7} + 196 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ao_ho_achl_vcm_afght |
| 59 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + T + 148 T^{2} + 849 T^{3} + 8431 T^{4} + 93089 T^{5} + 8431 p T^{6} + 849 p^{2} T^{7} + 148 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.59.b_fs_bgr_mmh_fhsj |
| 61 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 3 T + 115 T^{2} + 695 T^{3} + 8359 T^{4} + 71925 T^{5} + 8359 p T^{6} + 695 p^{2} T^{7} + 115 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.d_el_bat_mjn_eckj |
| 67 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 20 T + 418 T^{2} - 5384 T^{3} + 61489 T^{4} - 541919 T^{5} + 61489 p T^{6} - 5384 p^{2} T^{7} + 418 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.au_qc_ahzc_dmyz_abevrb |
| 71 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 25 T + 418 T^{2} - 5094 T^{3} + 51004 T^{4} - 453677 T^{5} + 51004 p T^{6} - 5094 p^{2} T^{7} + 418 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.az_qc_ahny_cxls_azvdd |
| 73 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - T + 130 T^{2} + 481 T^{3} + 10573 T^{4} + 52547 T^{5} + 10573 p T^{6} + 481 p^{2} T^{7} + 130 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.73.ab_fa_sn_pqr_cztb |
| 79 | $C_2 \times (C_2^4 : C_5)$ | \( 1 - 25 T + 425 T^{2} - 5267 T^{3} + 57465 T^{4} - 533887 T^{5} + 57465 p T^{6} - 5267 p^{2} T^{7} + 425 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.az_qj_ahup_dhaf_abejud |
| 83 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 37 T + 923 T^{2} + 15469 T^{3} + 204345 T^{4} + 2067001 T^{5} + 204345 p T^{6} + 15469 p^{2} T^{7} + 923 p^{3} T^{8} + 37 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.bl_bjn_wwz_lqhl_enpsb |
| 89 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 26 T + 612 T^{2} + 8954 T^{3} + 119683 T^{4} + 1176377 T^{5} + 119683 p T^{6} + 8954 p^{2} T^{7} + 612 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.ba_xo_ngk_gvbf_coyfh |
| 97 | $C_2 \times (C_2^4 : C_5)$ | \( 1 + 25 T + 592 T^{2} + 9091 T^{3} + 125239 T^{4} + 1293865 T^{5} + 125239 p T^{6} + 9091 p^{2} T^{7} + 592 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.z_wu_nlr_hdgx_cvqab |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.17280733709246060836855301252, −5.12006774682163757103608898624, −5.05423881490552695866729480300, −4.80659884982767636942799578738, −4.27979029861938361917501293435, −4.08352244913524903398192226394, −3.88924486708232831818403539776, −3.81780252327767789070046622298, −3.79551154706784116165004049832, −3.70976212545852526758348986625, −3.24953067978624564822216994053, −3.11967342462475199582596780172, −2.86420471644601322227472770795, −2.79816005065793480346261128281, −2.66078239946260571101234800893, −2.37191894092211948484486684400, −2.27030238106152733381097586588, −2.21303176729799453359569031737, −2.07447958437918420569067493572, −2.04035505736985434721231050506, −1.33858662021424176294426479901, −1.31358030648230184675655722136, −1.08148091791165143307343772969, −1.01626655309432496479837385540, −0.842749203974657005397904537714, 0, 0, 0, 0, 0,
0.842749203974657005397904537714, 1.01626655309432496479837385540, 1.08148091791165143307343772969, 1.31358030648230184675655722136, 1.33858662021424176294426479901, 2.04035505736985434721231050506, 2.07447958437918420569067493572, 2.21303176729799453359569031737, 2.27030238106152733381097586588, 2.37191894092211948484486684400, 2.66078239946260571101234800893, 2.79816005065793480346261128281, 2.86420471644601322227472770795, 3.11967342462475199582596780172, 3.24953067978624564822216994053, 3.70976212545852526758348986625, 3.79551154706784116165004049832, 3.81780252327767789070046622298, 3.88924486708232831818403539776, 4.08352244913524903398192226394, 4.27979029861938361917501293435, 4.80659884982767636942799578738, 5.05423881490552695866729480300, 5.12006774682163757103608898624, 5.17280733709246060836855301252
