| L(s) = 1 | − 5·5-s + 2·7-s − 9-s + 11-s + 4·13-s + 4·17-s − 7·19-s + 5·23-s + 15·25-s − 27-s + 4·29-s − 19·31-s − 10·35-s + 15·37-s + 25·41-s + 5·45-s + 11·47-s − 3·49-s + 3·53-s − 5·55-s + 59-s − 5·61-s − 2·63-s − 20·65-s − 9·67-s − 71-s − 73-s + ⋯ |
| L(s) = 1 | − 2.23·5-s + 0.755·7-s − 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.970·17-s − 1.60·19-s + 1.04·23-s + 3·25-s − 0.192·27-s + 0.742·29-s − 3.41·31-s − 1.69·35-s + 2.46·37-s + 3.90·41-s + 0.745·45-s + 1.60·47-s − 3/7·49-s + 0.412·53-s − 0.674·55-s + 0.130·59-s − 0.640·61-s − 0.251·63-s − 2.48·65-s − 1.09·67-s − 0.118·71-s − 0.117·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.106954725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.106954725\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 23 | $C_1$ | \( ( 1 - T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 + T^{2} + T^{3} - 4 T^{4} - 10 T^{5} - 4 p T^{6} + p^{2} T^{7} + p^{3} T^{8} + p^{5} T^{10} \) | 5.3.a_b_b_ae_ak |
| 7 | $C_2 \wr S_5$ | \( 1 - 2 T + p T^{2} + T^{3} + 30 T^{4} - 46 T^{5} + 30 p T^{6} + p^{2} T^{7} + p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.ac_h_b_be_abu |
| 11 | $C_2 \wr S_5$ | \( 1 - T + 20 T^{2} - 16 T^{3} + 227 T^{4} - 174 T^{5} + 227 p T^{6} - 16 p^{2} T^{7} + 20 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.11.ab_u_aq_it_ags |
| 13 | $C_2 \wr S_5$ | \( 1 - 4 T + 19 T^{2} - 133 T^{3} + 496 T^{4} - 1606 T^{5} + 496 p T^{6} - 133 p^{2} T^{7} + 19 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ae_t_afd_tc_acju |
| 17 | $C_2 \wr S_5$ | \( 1 - 4 T + 39 T^{2} - 115 T^{3} + 406 T^{4} - 1566 T^{5} + 406 p T^{6} - 115 p^{2} T^{7} + 39 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.ae_bn_ael_pq_acig |
| 19 | $C_2 \wr S_5$ | \( 1 + 7 T + 54 T^{2} + 352 T^{3} + 1949 T^{4} + 7810 T^{5} + 1949 p T^{6} + 352 p^{2} T^{7} + 54 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.h_cc_no_cwz_lok |
| 29 | $C_2 \wr S_5$ | \( 1 - 4 T + 104 T^{2} - 500 T^{3} + 4907 T^{4} - 22264 T^{5} + 4907 p T^{6} - 500 p^{2} T^{7} + 104 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.ae_ea_atg_hgt_abgyi |
| 31 | $C_2 \wr S_5$ | \( 1 + 19 T + 227 T^{2} + 2173 T^{3} + 16253 T^{4} + 98336 T^{5} + 16253 p T^{6} + 2173 p^{2} T^{7} + 227 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.t_it_dfp_ybd_fpme |
| 37 | $C_2 \wr S_5$ | \( 1 - 15 T + 249 T^{2} - 2256 T^{3} + 20618 T^{4} - 125810 T^{5} + 20618 p T^{6} - 2256 p^{2} T^{7} + 249 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.ap_jp_adiu_bena_ahecw |
| 41 | $C_2 \wr S_5$ | \( 1 - 25 T + 417 T^{2} - 4753 T^{3} + 42913 T^{4} - 303514 T^{5} + 42913 p T^{6} - 4753 p^{2} T^{7} + 417 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.az_qb_ahav_clmn_argzq |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{5} \) | 5.43.a_ih_a_bbje_a |
| 47 | $C_2 \wr S_5$ | \( 1 - 11 T + 125 T^{2} - 952 T^{3} + 5780 T^{4} - 41402 T^{5} + 5780 p T^{6} - 952 p^{2} T^{7} + 125 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.al_ev_abkq_ioi_acjgk |
| 53 | $C_2 \wr S_5$ | \( 1 - 3 T + 105 T^{2} - 196 T^{3} + 8458 T^{4} - 24194 T^{5} + 8458 p T^{6} - 196 p^{2} T^{7} + 105 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ad_eb_aho_mni_abjuo |
| 59 | $C_2 \wr S_5$ | \( 1 - T + 153 T^{2} + 14236 T^{4} - 6606 T^{5} + 14236 p T^{6} + 153 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.59.ab_fx_a_vbo_ajuc |
| 61 | $C_2 \wr S_5$ | \( 1 + 5 T + 190 T^{2} + 652 T^{3} + 18257 T^{4} + 49998 T^{5} + 18257 p T^{6} + 652 p^{2} T^{7} + 190 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.f_hi_zc_bbaf_cvza |
| 67 | $C_2 \wr S_5$ | \( 1 + 9 T + 209 T^{2} + 1980 T^{3} + 24396 T^{4} + 176326 T^{5} + 24396 p T^{6} + 1980 p^{2} T^{7} + 209 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.j_ib_cye_bkci_kavu |
| 71 | $C_2 \wr S_5$ | \( 1 + T + 141 T^{2} - 123 T^{3} + 12967 T^{4} - 23580 T^{5} + 12967 p T^{6} - 123 p^{2} T^{7} + 141 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.71.b_fl_aet_tet_abiwy |
| 73 | $C_2 \wr S_5$ | \( 1 + T + 207 T^{2} + 20 T^{3} + 20760 T^{4} - 9066 T^{5} + 20760 p T^{6} + 20 p^{2} T^{7} + 207 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.73.b_hz_u_besm_anks |
| 79 | $C_2 \wr S_5$ | \( 1 - 2 T + 267 T^{2} - 760 T^{3} + 34506 T^{4} - 94092 T^{5} + 34506 p T^{6} - 760 p^{2} T^{7} + 267 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.ac_kh_abdg_bzbe_afjey |
| 83 | $C_2 \wr S_5$ | \( 1 - 45 T + 1199 T^{2} - 21528 T^{3} + 290714 T^{4} - 2994854 T^{5} + 290714 p T^{6} - 21528 p^{2} T^{7} + 1199 p^{3} T^{8} - 45 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.abt_bud_abfwa_qobi_agokgs |
| 89 | $C_2 \wr S_5$ | \( 1 - 6 T + 309 T^{2} - 1624 T^{3} + 45458 T^{4} - 202212 T^{5} + 45458 p T^{6} - 1624 p^{2} T^{7} + 309 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.ag_lx_ackm_cpgk_alndk |
| 97 | $C_2 \wr S_5$ | \( 1 - 25 T + 446 T^{2} - 5536 T^{3} + 65833 T^{4} - 653150 T^{5} + 65833 p T^{6} - 5536 p^{2} T^{7} + 446 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.az_re_aiey_dtkb_ablefe |
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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
−5.45116032660235916260362165796, −5.32014967978384418683299392838, −5.10963416088376515193759381939, −5.09825589430655363909642751251, −4.69648518855496177965789033033, −4.64863388003871234103167066276, −4.37295117066608871567554175610, −4.26667838216577478929509573491, −4.10840656914172198275122457122, −3.93720614406734901642418798601, −3.78412579682264420598315908340, −3.50498762785343705309704084704, −3.39846428044004453365231618992, −3.25875493881566695097223488837, −3.15386255094323939197696483215, −2.54526778852463296582397838950, −2.40917653730055886239035235130, −2.37603137320533751218488741797, −2.24996613830601281615029504073, −1.71606638558687527666591548000, −1.46283804703528930027940476065, −1.09845870033423592219487773786, −0.833169813124345125532751451384, −0.73930114888484845428126629533, −0.32957100908932285842405515070,
0.32957100908932285842405515070, 0.73930114888484845428126629533, 0.833169813124345125532751451384, 1.09845870033423592219487773786, 1.46283804703528930027940476065, 1.71606638558687527666591548000, 2.24996613830601281615029504073, 2.37603137320533751218488741797, 2.40917653730055886239035235130, 2.54526778852463296582397838950, 3.15386255094323939197696483215, 3.25875493881566695097223488837, 3.39846428044004453365231618992, 3.50498762785343705309704084704, 3.78412579682264420598315908340, 3.93720614406734901642418798601, 4.10840656914172198275122457122, 4.26667838216577478929509573491, 4.37295117066608871567554175610, 4.64863388003871234103167066276, 4.69648518855496177965789033033, 5.09825589430655363909642751251, 5.10963416088376515193759381939, 5.32014967978384418683299392838, 5.45116032660235916260362165796
