| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 3·8-s − 4·9-s + 4·11-s + 12-s − 3·13-s − 14-s + 4·16-s + 5·17-s + 4·18-s + 6·19-s + 21-s − 4·22-s + 4·23-s − 3·24-s + 3·26-s − 7·27-s + 28-s + 2·29-s + 21·31-s − 2·32-s + 4·33-s − 5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s − 4/3·9-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 16-s + 1.21·17-s + 0.942·18-s + 1.37·19-s + 0.218·21-s − 0.852·22-s + 0.834·23-s − 0.612·24-s + 0.588·26-s − 1.34·27-s + 0.188·28-s + 0.371·29-s + 3.77·31-s − 0.353·32-s + 0.696·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.758877838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.758877838\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 - T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + T + p T^{3} + T^{4} - 3 T^{5} + p T^{6} + p^{3} T^{7} + p^{4} T^{9} + p^{5} T^{10} \) | 5.2.b_a_c_b_ad |
| 3 | $C_2 \wr S_5$ | \( 1 - T + 5 T^{2} - 2 T^{3} + 23 T^{4} - 19 T^{5} + 23 p T^{6} - 2 p^{2} T^{7} + 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.3.ab_f_ac_x_at |
| 7 | $C_2 \wr S_5$ | \( 1 - T + 13 T^{2} - 26 T^{3} + 137 T^{4} - 169 T^{5} + 137 p T^{6} - 26 p^{2} T^{7} + 13 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.7.ab_n_aba_fh_agn |
| 11 | $C_2 \wr S_5$ | \( 1 - 4 T + 3 p T^{2} - 56 T^{3} + 328 T^{4} - 204 T^{5} + 328 p T^{6} - 56 p^{2} T^{7} + 3 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.ae_bh_ace_mq_ahw |
| 13 | $C_2 \wr S_5$ | \( 1 + 3 T + 3 p T^{2} + 98 T^{3} + 853 T^{4} + 1761 T^{5} + 853 p T^{6} + 98 p^{2} T^{7} + 3 p^{4} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.d_bn_du_bgv_cpt |
| 19 | $C_2 \wr S_5$ | \( 1 - 6 T + 63 T^{2} - 8 p T^{3} + 1114 T^{4} - 1044 T^{5} + 1114 p T^{6} - 8 p^{3} T^{7} + 63 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.ag_cl_afw_bqw_aboe |
| 23 | $C_2 \wr S_5$ | \( 1 - 4 T + 3 p T^{2} - 116 T^{3} + 1792 T^{4} - 996 T^{5} + 1792 p T^{6} - 116 p^{2} T^{7} + 3 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.ae_cr_aem_cqy_abmi |
| 29 | $C_2 \wr S_5$ | \( 1 - 2 T + 69 T^{2} - 184 T^{3} + 3142 T^{4} - 7068 T^{5} + 3142 p T^{6} - 184 p^{2} T^{7} + 69 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.ac_cr_ahc_eqw_aklw |
| 31 | $C_2 \wr S_5$ | \( 1 - 21 T + 257 T^{2} - 2246 T^{3} + 16111 T^{4} - 96739 T^{5} + 16111 p T^{6} - 2246 p^{2} T^{7} + 257 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.av_jx_adik_xvr_afnct |
| 37 | $C_2 \wr S_5$ | \( 1 + 2 T + 105 T^{2} + 40 T^{3} + 5930 T^{4} + 1308 T^{5} + 5930 p T^{6} + 40 p^{2} T^{7} + 105 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.c_eb_bo_iuc_byi |
| 41 | $C_2 \wr S_5$ | \( 1 + 8 T + 185 T^{2} + 1144 T^{3} + 14302 T^{4} + 66960 T^{5} + 14302 p T^{6} + 1144 p^{2} T^{7} + 185 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.i_hd_bsa_vec_dvbk |
| 43 | $C_2 \wr S_5$ | \( 1 + 4 T + 79 T^{2} + 192 T^{3} + 4818 T^{4} + 16424 T^{5} + 4818 p T^{6} + 192 p^{2} T^{7} + 79 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.e_db_hk_hdi_yhs |
| 47 | $C_2 \wr S_5$ | \( 1 + 2 T + 147 T^{2} - 104 T^{3} + 9154 T^{4} - 18132 T^{5} + 9154 p T^{6} - 104 p^{2} T^{7} + 147 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.c_fr_aea_noc_abavk |
| 53 | $C_2 \wr S_5$ | \( 1 - 21 T + 419 T^{2} - 4902 T^{3} + 52981 T^{4} - 401643 T^{5} + 52981 p T^{6} - 4902 p^{2} T^{7} + 419 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.av_qd_ahgo_dajt_awwdv |
| 59 | $C_2 \wr S_5$ | \( 1 - 12 T + 179 T^{2} - 1560 T^{3} + 16294 T^{4} - 112296 T^{5} + 16294 p T^{6} - 1560 p^{2} T^{7} + 179 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.am_gx_acia_ycs_agkdc |
| 61 | $C_2 \wr S_5$ | \( 1 + 2 T + 265 T^{2} + 408 T^{3} + 30258 T^{4} + 35692 T^{5} + 30258 p T^{6} + 408 p^{2} T^{7} + 265 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.c_kf_ps_bstu_cauu |
| 67 | $C_2 \wr S_5$ | \( 1 + 12 T + 159 T^{2} + 1808 T^{3} + 17386 T^{4} + 161544 T^{5} + 17386 p T^{6} + 1808 p^{2} T^{7} + 159 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.m_gd_cro_zss_jezg |
| 71 | $C_2 \wr S_5$ | \( 1 - 21 T + 337 T^{2} - 3474 T^{3} + 33841 T^{4} - 269733 T^{5} + 33841 p T^{6} - 3474 p^{2} T^{7} + 337 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.av_mz_afdq_bybp_apjaj |
| 73 | $C_2 \wr S_5$ | \( 1 - 22 T + 301 T^{2} - 2816 T^{3} + 25754 T^{4} - 218404 T^{5} + 25754 p T^{6} - 2816 p^{2} T^{7} + 301 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.aw_lp_aeei_bmco_amlce |
| 79 | $C_2 \wr S_5$ | \( 1 - 41 T + 919 T^{2} - 14404 T^{3} + 175265 T^{4} - 1721495 T^{5} + 175265 p T^{6} - 14404 p^{2} T^{7} + 919 p^{3} T^{8} - 41 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.abp_bjj_avia_jzgz_adtypj |
| 83 | $C_2 \wr S_5$ | \( 1 + 8 T + 351 T^{2} + 2176 T^{3} + 53722 T^{4} + 256944 T^{5} + 53722 p T^{6} + 2176 p^{2} T^{7} + 351 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.i_nn_dfs_dbmg_oqcm |
| 89 | $C_2 \wr S_5$ | \( 1 + 10 T + 201 T^{2} + 2336 T^{3} + 25582 T^{4} + 239388 T^{5} + 25582 p T^{6} + 2336 p^{2} T^{7} + 201 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.k_ht_dlw_blvy_nqdg |
| 97 | $C_2 \wr S_5$ | \( 1 + 20 T + 541 T^{2} + 6880 T^{3} + 104786 T^{4} + 950360 T^{5} + 104786 p T^{6} + 6880 p^{2} T^{7} + 541 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.u_uv_keq_fzag_ccbwi |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.81733698517595323537860173464, −6.80907282210782151573627548617, −6.51572836415958185428539607920, −6.25145415343839697720964370154, −6.18583462385291200260309802812, −6.06137777324432009059078310838, −5.45538314786289126380052227641, −5.32742209612879555683773969352, −5.27043275954358203445584930202, −5.21395535328633499078677321425, −4.90133776319090147625981258287, −4.47221018802310014185416387201, −4.20926726214541565482767402750, −4.14407748773180695389941393971, −3.49839692802170015820121975289, −3.33823682520123434656541382227, −3.31201376684038709815969091525, −3.17966806370995735297380510914, −2.66845775379697339868635775421, −2.43994207376998844318899381848, −2.33654851519484208365600763619, −1.84188267605493801351100008422, −1.31759348064657741037092219195, −0.861447237802358871060367459182, −0.73606573268920149290747172697,
0.73606573268920149290747172697, 0.861447237802358871060367459182, 1.31759348064657741037092219195, 1.84188267605493801351100008422, 2.33654851519484208365600763619, 2.43994207376998844318899381848, 2.66845775379697339868635775421, 3.17966806370995735297380510914, 3.31201376684038709815969091525, 3.33823682520123434656541382227, 3.49839692802170015820121975289, 4.14407748773180695389941393971, 4.20926726214541565482767402750, 4.47221018802310014185416387201, 4.90133776319090147625981258287, 5.21395535328633499078677321425, 5.27043275954358203445584930202, 5.32742209612879555683773969352, 5.45538314786289126380052227641, 6.06137777324432009059078310838, 6.18583462385291200260309802812, 6.25145415343839697720964370154, 6.51572836415958185428539607920, 6.80907282210782151573627548617, 6.81733698517595323537860173464
