| L(s) = 1 | + 2-s + 5·3-s − 4-s + 5·6-s − 4·7-s − 2·8-s + 15·9-s − 5·12-s + 4·13-s − 4·14-s + 8·17-s + 15·18-s − 6·19-s − 20·21-s − 9·23-s − 10·24-s + 4·26-s + 35·27-s + 4·28-s − 2·29-s − 3·31-s + 4·32-s + 8·34-s − 15·36-s + 37-s − 6·38-s + 20·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.88·3-s − 1/2·4-s + 2.04·6-s − 1.51·7-s − 0.707·8-s + 5·9-s − 1.44·12-s + 1.10·13-s − 1.06·14-s + 1.94·17-s + 3.53·18-s − 1.37·19-s − 4.36·21-s − 1.87·23-s − 2.04·24-s + 0.784·26-s + 6.73·27-s + 0.755·28-s − 0.371·29-s − 0.538·31-s + 0.707·32-s + 1.37·34-s − 5/2·36-s + 0.164·37-s − 0.973·38-s + 3.20·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(36.08857899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.08857899\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | $C_1$ | \( ( 1 - T )^{5} \) | |
| 5 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T + p T^{2} - T^{3} + T^{4} - p T^{5} + p T^{6} - p^{2} T^{7} + p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.2.ab_c_ab_b_ac |
| 7 | $C_2 \wr S_5$ | \( 1 + 4 T + 17 T^{2} + 5 p T^{3} + 80 T^{4} + 130 T^{5} + 80 p T^{6} + 5 p^{3} T^{7} + 17 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.e_r_bj_dc_fa |
| 13 | $C_2 \wr S_5$ | \( 1 - 4 T + 28 T^{2} - 157 T^{3} + 711 T^{4} - 2221 T^{5} + 711 p T^{6} - 157 p^{2} T^{7} + 28 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ae_bc_agb_bbj_adhl |
| 17 | $C_2 \wr S_5$ | \( 1 - 8 T + 78 T^{2} - 414 T^{3} + 2425 T^{4} - 9668 T^{5} + 2425 p T^{6} - 414 p^{2} T^{7} + 78 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.ai_da_apy_dph_aohw |
| 19 | $C_2 \wr S_5$ | \( 1 + 6 T + 31 T^{2} + 78 T^{3} - 79 T^{4} - 124 T^{5} - 79 p T^{6} + 78 p^{2} T^{7} + 31 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.g_bf_da_adb_aeu |
| 23 | $C_2 \wr S_5$ | \( 1 + 9 T + 102 T^{2} + 592 T^{3} + 4081 T^{4} + 17614 T^{5} + 4081 p T^{6} + 592 p^{2} T^{7} + 102 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.j_dy_wu_gaz_babm |
| 29 | $C_2 \wr S_5$ | \( 1 + 2 T + 70 T^{2} + 140 T^{3} + 3037 T^{4} + 5524 T^{5} + 3037 p T^{6} + 140 p^{2} T^{7} + 70 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.c_cs_fk_emv_iem |
| 31 | $C_2 \wr S_5$ | \( 1 + 3 T + 27 T^{2} + 287 T^{3} + 1947 T^{4} + 2988 T^{5} + 1947 p T^{6} + 287 p^{2} T^{7} + 27 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.d_bb_lb_cwx_eky |
| 37 | $C_2 \wr S_5$ | \( 1 - T + 84 T^{2} + 73 T^{3} + 2591 T^{4} + 8124 T^{5} + 2591 p T^{6} + 73 p^{2} T^{7} + 84 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.37.ab_dg_cv_dvr_mam |
| 41 | $C_2 \wr S_5$ | \( 1 + T + 48 T^{2} - 120 T^{3} + 3715 T^{4} + 2830 T^{5} + 3715 p T^{6} - 120 p^{2} T^{7} + 48 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.41.b_bw_aeq_fmx_eew |
| 43 | $C_2 \wr S_5$ | \( 1 - 7 T + 158 T^{2} - 909 T^{3} + 11893 T^{4} - 54044 T^{5} + 11893 p T^{6} - 909 p^{2} T^{7} + 158 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.ah_gc_abiz_rpl_adbyq |
| 47 | $C_2 \wr S_5$ | \( 1 + 14 T + 179 T^{2} + 1136 T^{3} + 8674 T^{4} + 43396 T^{5} + 8674 p T^{6} + 1136 p^{2} T^{7} + 179 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.o_gx_brs_mvq_cmfc |
| 53 | $C_2 \wr S_5$ | \( 1 - 3 T + 168 T^{2} - 298 T^{3} + 14851 T^{4} - 22846 T^{5} + 14851 p T^{6} - 298 p^{2} T^{7} + 168 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ad_gm_alm_vzf_abhus |
| 59 | $C_2 \wr S_5$ | \( 1 - 18 T + 228 T^{2} - 1996 T^{3} + 15775 T^{4} - 110692 T^{5} + 15775 p T^{6} - 1996 p^{2} T^{7} + 228 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.as_iu_acyu_xit_aghtk |
| 61 | $C_2 \wr S_5$ | \( 1 - 2 T + 170 T^{2} + 536 T^{3} + 10121 T^{4} + 81628 T^{5} + 10121 p T^{6} + 536 p^{2} T^{7} + 170 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.ac_go_uq_ozh_eqto |
| 67 | $C_2 \wr S_5$ | \( 1 - 12 T + 272 T^{2} - 2298 T^{3} + 33047 T^{4} - 217300 T^{5} + 33047 p T^{6} - 2298 p^{2} T^{7} + 272 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.am_km_adkk_bwxb_amjls |
| 71 | $C_2 \wr S_5$ | \( 1 - 8 T + 128 T^{2} + 82 T^{3} - 797 T^{4} + 76700 T^{5} - 797 p T^{6} + 82 p^{2} T^{7} + 128 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.ai_ey_de_aber_ejma |
| 73 | $C_2 \wr S_5$ | \( 1 - 16 T + 206 T^{2} - 327 T^{3} - 7937 T^{4} + 170293 T^{5} - 7937 p T^{6} - 327 p^{2} T^{7} + 206 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.aq_hy_amp_alth_jrxt |
| 79 | $C_2 \wr S_5$ | \( 1 - 11 T + 392 T^{2} - 3173 T^{3} + 60757 T^{4} - 363188 T^{5} + 60757 p T^{6} - 3173 p^{2} T^{7} + 392 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.al_pc_aesb_dlwv_aurgu |
| 83 | $C_2 \wr S_5$ | \( 1 + 39 T + 838 T^{2} + 12626 T^{3} + 149329 T^{4} + 1469246 T^{5} + 149329 p T^{6} + 12626 p^{2} T^{7} + 838 p^{3} T^{8} + 39 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.bn_bgg_srq_imxl_dfplm |
| 89 | $C_2 \wr S_5$ | \( 1 + T + 288 T^{2} + 72 T^{3} + 43507 T^{4} + 13006 T^{5} + 43507 p T^{6} + 72 p^{2} T^{7} + 288 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.89.b_lc_cu_cmjj_tgg |
| 97 | $C_2 \wr S_5$ | \( 1 - 11 T + 391 T^{2} - 3044 T^{3} + 64584 T^{4} - 385538 T^{5} + 64584 p T^{6} - 3044 p^{2} T^{7} + 391 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.al_pb_aenc_droa_avyik |
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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
−4.32527794785154893215265953667, −4.14205532698438379937512136417, −4.11103121128900485019435349789, −3.95580639537453201141923456488, −3.91958777873014298539652300139, −3.69238426530140799910855120295, −3.54006562359454372663432092454, −3.50731513507856996867281622586, −3.35081229632577392906050619398, −3.22502061510829257342458249641, −3.12660769093351000333326575646, −2.84051954141720492902713193753, −2.61442776198897631258262146151, −2.60585517363630114074351433850, −2.31835664849162140087813017653, −2.25647162031856823304251312295, −1.94833178360694119095618507142, −1.86268721798963396922812445623, −1.71565058165657843038549427797, −1.41865528958796829654820074672, −1.36702384975606235619616512680, −0.794037486301738826348798077294, −0.78796317335720383787200029942, −0.49357687902542137785722522708, −0.35228299519619646760275372912,
0.35228299519619646760275372912, 0.49357687902542137785722522708, 0.78796317335720383787200029942, 0.794037486301738826348798077294, 1.36702384975606235619616512680, 1.41865528958796829654820074672, 1.71565058165657843038549427797, 1.86268721798963396922812445623, 1.94833178360694119095618507142, 2.25647162031856823304251312295, 2.31835664849162140087813017653, 2.60585517363630114074351433850, 2.61442776198897631258262146151, 2.84051954141720492902713193753, 3.12660769093351000333326575646, 3.22502061510829257342458249641, 3.35081229632577392906050619398, 3.50731513507856996867281622586, 3.54006562359454372663432092454, 3.69238426530140799910855120295, 3.91958777873014298539652300139, 3.95580639537453201141923456488, 4.11103121128900485019435349789, 4.14205532698438379937512136417, 4.32527794785154893215265953667
