| L(s) = 1 | + 7·4-s + 6·5-s − 6·7-s + 13·9-s − 2·13-s + 23·16-s + 42·20-s − 14·23-s + 5·25-s − 42·28-s − 36·35-s + 91·36-s + 78·45-s − 7·49-s − 14·52-s − 18·53-s − 56·59-s − 78·63-s + 42·64-s − 12·65-s − 26·67-s + 42·71-s + 138·80-s + 88·81-s − 18·83-s + 12·91-s − 98·92-s + ⋯ |
| L(s) = 1 | + 7/2·4-s + 2.68·5-s − 2.26·7-s + 13/3·9-s − 0.554·13-s + 23/4·16-s + 9.39·20-s − 2.91·23-s + 25-s − 7.93·28-s − 6.08·35-s + 91/6·36-s + 11.6·45-s − 49-s − 1.94·52-s − 2.47·53-s − 7.29·59-s − 9.82·63-s + 21/4·64-s − 1.48·65-s − 3.17·67-s + 4.98·71-s + 15.4·80-s + 88/9·81-s − 1.97·83-s + 1.25·91-s − 10.2·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5123064860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5123064860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 29 | \( 1 \) | |
| good | 2 | \( 1 - 7 T^{2} + 13 p T^{4} - 63 T^{6} + 13 p^{3} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) | 6.2.a_ah_a_ba_a_acl |
| 3 | \( 1 - 13 T^{2} + p^{4} T^{4} - 305 T^{6} + p^{6} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \) | 6.3.a_an_a_dd_a_alt |
| 5 | \( ( 1 - 3 T + 11 T^{2} - 31 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.5.ag_bf_aey_qb_abss_egl |
| 7 | \( ( 1 + 3 T + 17 T^{2} + 43 T^{3} + 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.7.g_br_hg_bef_dra_lbb |
| 11 | \( 1 - 25 T^{2} + 485 T^{4} - 5601 T^{6} + 485 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) | 6.11.a_az_a_sr_a_aihl |
| 13 | \( ( 1 + T + 9 T^{2} - 17 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.13.c_t_aq_kv_kg_kod |
| 17 | \( 1 - 78 T^{2} + 2783 T^{4} - 59300 T^{6} + 2783 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) | 6.17.a_ada_a_edb_a_adjsu |
| 19 | \( 1 - 97 T^{2} + 4217 T^{4} - 103761 T^{6} + 4217 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) | 6.19.a_adt_a_ggf_a_afxmv |
| 23 | \( ( 1 + 7 T + 3 p T^{2} + 273 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.23.o_hf_cge_rkf_dvok_vaxr |
| 31 | \( 1 - 97 T^{2} + 5513 T^{4} - 208065 T^{6} + 5513 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) | 6.31.a_adt_a_ieb_a_alvun |
| 37 | \( 1 - 181 T^{2} + 14941 T^{4} - 709681 T^{6} + 14941 p^{2} T^{8} - 181 p^{4} T^{10} + p^{6} T^{12} \) | 6.37.a_agz_a_wcr_a_abojvl |
| 41 | \( 1 - 194 T^{2} + 17103 T^{4} - 887996 T^{6} + 17103 p^{2} T^{8} - 194 p^{4} T^{10} + p^{6} T^{12} \) | 6.41.a_ahm_a_zhv_a_abynps |
| 43 | \( 1 - 213 T^{2} + 20397 T^{4} - 1125313 T^{6} + 20397 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} \) | 6.43.a_aif_a_been_a_acmarh |
| 47 | \( 1 - 157 T^{2} + 14477 T^{4} - 817737 T^{6} + 14477 p^{2} T^{8} - 157 p^{4} T^{10} + p^{6} T^{12} \) | 6.47.a_agb_a_vkv_a_abunrl |
| 53 | \( ( 1 + 9 T + 179 T^{2} + 953 T^{3} + 179 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.53.s_qx_hpg_dwvx_bgacs_lbtid |
| 59 | \( ( 1 + 28 T + 429 T^{2} + 4032 T^{3} + 429 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.59.ce_cle_bvme_bafdv_lcoro_drymgq |
| 61 | \( 1 - 325 T^{2} + 45977 T^{4} - 3644337 T^{6} + 45977 p^{2} T^{8} - 325 p^{4} T^{10} + p^{6} T^{12} \) | 6.61.a_amn_a_cqaj_a_ahzjav |
| 67 | \( ( 1 + 13 T + 171 T^{2} + 1729 T^{3} + 171 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.67.ba_tr_lsa_fnqz_cfgba_ttowt |
| 71 | \( ( 1 - 21 T + 339 T^{2} - 3171 T^{3} + 339 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.71.abq_brb_abelo_qwfx_ahjwzu_cragxj |
| 73 | \( 1 - 257 T^{2} + 37357 T^{4} - 3317489 T^{6} + 37357 p^{2} T^{8} - 257 p^{4} T^{10} + p^{6} T^{12} \) | 6.73.a_ajx_a_cdgv_a_ahgtnt |
| 79 | \( 1 - 357 T^{2} + 58749 T^{4} - 5811617 T^{6} + 58749 p^{2} T^{8} - 357 p^{4} T^{10} + p^{6} T^{12} \) | 6.79.a_ant_a_dixp_a_amsrbt |
| 83 | \( ( 1 + 9 T + 227 T^{2} + 1325 T^{3} + 227 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.83.s_up_jzc_glgn_ciozq_bbndgh |
| 89 | \( 1 - 373 T^{2} + 65909 T^{4} - 7224393 T^{6} + 65909 p^{2} T^{8} - 373 p^{4} T^{10} + p^{6} T^{12} \) | 6.89.a_aoj_a_dtmz_a_apvazh |
| 97 | \( 1 - 313 T^{2} + 41953 T^{4} - 4074025 T^{6} + 41953 p^{2} T^{8} - 313 p^{4} T^{10} + p^{6} T^{12} \) | 6.97.a_amb_a_ckbp_a_aixurh |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.74734239089058889022773918832, −5.40478640231972228589078084433, −5.11703541094324348790220189404, −4.90607357316287714778865667313, −4.57954133317904443054422549116, −4.55433679383011011343857713984, −4.49898816108234616844708057071, −4.44195123470968051416908931178, −3.90292686737028882998879577419, −3.77669367836919780328807930968, −3.66015231362953174897310148166, −3.56251006901415576557139193584, −3.10878710753266095419198814711, −2.96335998085194522442243742214, −2.86129923982311047396299629010, −2.81352639158591914194015902355, −2.38984809545065968487931630585, −2.10446250264850706279372336461, −1.87350931284075803655797501416, −1.84499959060716781780780313188, −1.83082706879780840088931908605, −1.52237732585436772160549528942, −1.37131935278624560137010563847, −1.35547734050390758203242567585, −0.06112259549857286529746447243,
0.06112259549857286529746447243, 1.35547734050390758203242567585, 1.37131935278624560137010563847, 1.52237732585436772160549528942, 1.83082706879780840088931908605, 1.84499959060716781780780313188, 1.87350931284075803655797501416, 2.10446250264850706279372336461, 2.38984809545065968487931630585, 2.81352639158591914194015902355, 2.86129923982311047396299629010, 2.96335998085194522442243742214, 3.10878710753266095419198814711, 3.56251006901415576557139193584, 3.66015231362953174897310148166, 3.77669367836919780328807930968, 3.90292686737028882998879577419, 4.44195123470968051416908931178, 4.49898816108234616844708057071, 4.55433679383011011343857713984, 4.57954133317904443054422549116, 4.90607357316287714778865667313, 5.11703541094324348790220189404, 5.40478640231972228589078084433, 5.74734239089058889022773918832
Plot not available for L-functions of degree greater than 10.
