| L(s) = 1 | − 4·3-s − 5·5-s + 4·7-s + 4·9-s − 2·11-s − 4·13-s + 20·15-s − 12·17-s + 5·19-s − 16·21-s + 8·23-s + 15·25-s + 4·27-s − 6·29-s + 10·31-s + 8·33-s − 20·35-s − 6·37-s + 16·39-s − 8·41-s − 12·43-s − 20·45-s + 16·47-s − 6·49-s + 48·51-s − 18·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2.23·5-s + 1.51·7-s + 4/3·9-s − 0.603·11-s − 1.10·13-s + 5.16·15-s − 2.91·17-s + 1.14·19-s − 3.49·21-s + 1.66·23-s + 3·25-s + 0.769·27-s − 1.11·29-s + 1.79·31-s + 1.39·33-s − 3.38·35-s − 0.986·37-s + 2.56·39-s − 1.24·41-s − 1.82·43-s − 2.98·45-s + 2.33·47-s − 6/7·49-s + 6.72·51-s − 2.47·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 4 T + 4 p T^{2} + 28 T^{3} + 55 T^{4} + 106 T^{5} + 55 p T^{6} + 28 p^{2} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.3.e_m_bc_cd_ec |
| 7 | $C_2 \wr S_5$ | \( 1 - 4 T + 22 T^{2} - 52 T^{3} + 181 T^{4} - 340 T^{5} + 181 p T^{6} - 52 p^{2} T^{7} + 22 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.ae_w_aca_gz_anc |
| 11 | $C_2 \wr S_5$ | \( 1 + 2 T + 23 T^{2} + 4 p T^{3} + 362 T^{4} + 420 T^{5} + 362 p T^{6} + 4 p^{3} T^{7} + 23 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.c_x_bs_ny_qe |
| 13 | $C_2 \wr S_5$ | \( 1 + 4 T + 46 T^{2} + 134 T^{3} + 895 T^{4} + 2134 T^{5} + 895 p T^{6} + 134 p^{2} T^{7} + 46 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.e_bu_fe_bil_dec |
| 17 | $C_2 \wr S_5$ | \( 1 + 12 T + 98 T^{2} + 514 T^{3} + 2349 T^{4} + 9348 T^{5} + 2349 p T^{6} + 514 p^{2} T^{7} + 98 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.m_du_tu_dmj_nvo |
| 23 | $C_2 \wr S_5$ | \( 1 - 8 T + 78 T^{2} - 584 T^{3} + 3493 T^{4} - 17628 T^{5} + 3493 p T^{6} - 584 p^{2} T^{7} + 78 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.ai_da_awm_fej_abaca |
| 29 | $C_2 \wr S_5$ | \( 1 + 6 T + 64 T^{2} + 106 T^{3} + 1655 T^{4} + 1536 T^{5} + 1655 p T^{6} + 106 p^{2} T^{7} + 64 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.g_cm_ec_clr_chc |
| 31 | $C_2 \wr S_5$ | \( 1 - 10 T + 115 T^{2} - 760 T^{3} + 5458 T^{4} - 29788 T^{5} + 5458 p T^{6} - 760 p^{2} T^{7} + 115 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.ak_el_abdg_iby_absbs |
| 37 | $C_2 \wr S_5$ | \( 1 + 6 T + 87 T^{2} + 526 T^{3} + 5556 T^{4} + 24712 T^{5} + 5556 p T^{6} + 526 p^{2} T^{7} + 87 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.g_dj_ug_ifs_bkom |
| 41 | $C_2 \wr S_5$ | \( 1 + 8 T + 193 T^{2} + 1152 T^{3} + 15206 T^{4} + 67888 T^{5} + 15206 p T^{6} + 1152 p^{2} T^{7} + 193 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.i_hl_bsi_wmw_dwlc |
| 43 | $C_2 \wr S_5$ | \( 1 + 12 T + 107 T^{2} + 1148 T^{3} + 8942 T^{4} + 53024 T^{5} + 8942 p T^{6} + 1148 p^{2} T^{7} + 107 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.m_ed_bse_nfy_dalk |
| 47 | $C_2 \wr S_5$ | \( 1 - 16 T + 255 T^{2} - 2204 T^{3} + 20430 T^{4} - 129896 T^{5} + 20430 p T^{6} - 2204 p^{2} T^{7} + 255 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.aq_jv_adgu_befu_ahkea |
| 53 | $C_2 \wr S_5$ | \( 1 + 18 T + 186 T^{2} + 1916 T^{3} + 17247 T^{4} + 130458 T^{5} + 17247 p T^{6} + 1916 p^{2} T^{7} + 186 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.s_he_cvs_znj_hkzq |
| 59 | $C_2 \wr S_5$ | \( 1 + 8 T + 240 T^{2} + 1750 T^{3} + 25327 T^{4} + 151196 T^{5} + 25327 p T^{6} + 1750 p^{2} T^{7} + 240 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.i_jg_cpi_blmd_iprg |
| 61 | $C_2 \wr S_5$ | \( 1 - 2 T + 277 T^{2} - 508 T^{3} + 32150 T^{4} - 47028 T^{5} + 32150 p T^{6} - 508 p^{2} T^{7} + 277 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.ac_kr_ato_bvoo_acrou |
| 67 | $C_2 \wr S_5$ | \( 1 + 10 T + 256 T^{2} + 2166 T^{3} + 28507 T^{4} + 200414 T^{5} + 28507 p T^{6} + 2166 p^{2} T^{7} + 256 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.k_jw_dfi_bqel_lkmg |
| 71 | $C_2 \wr S_5$ | \( 1 + 18 T + 159 T^{2} + 1024 T^{3} + 2774 T^{4} - 14052 T^{5} + 2774 p T^{6} + 1024 p^{2} T^{7} + 159 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.s_gd_bnk_ecs_auum |
| 73 | $C_2 \wr S_5$ | \( 1 + 28 T + 618 T^{2} + 8822 T^{3} + 107301 T^{4} + 984108 T^{5} + 107301 p T^{6} + 8822 p^{2} T^{7} + 618 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.bc_xu_nbi_gcsz_cdzui |
| 79 | $C_2 \wr S_5$ | \( 1 - 14 T + 355 T^{2} - 3416 T^{3} + 50770 T^{4} - 368276 T^{5} + 50770 p T^{6} - 3416 p^{2} T^{7} + 355 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.ao_nr_afbk_cxcs_auyum |
| 83 | $C_2 \wr S_5$ | \( 1 + 8 T + 211 T^{2} + 2484 T^{3} + 23022 T^{4} + 296280 T^{5} + 23022 p T^{6} + 2484 p^{2} T^{7} + 211 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.i_id_dro_bibm_qwhk |
| 89 | $C_2 \wr S_5$ | \( 1 + 30 T + 677 T^{2} + 10744 T^{3} + 140338 T^{4} + 1436436 T^{5} + 140338 p T^{6} + 10744 p^{2} T^{7} + 677 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.be_bab_pxg_hzpq_ddsxo |
| 97 | $C_2 \wr S_5$ | \( 1 + 18 T + 359 T^{2} + 4250 T^{3} + 60944 T^{4} + 586328 T^{5} + 60944 p T^{6} + 4250 p^{2} T^{7} + 359 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.s_nv_ghm_dmea_bhjjc |
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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
−5.41428776650590569393751473760, −5.33706283337834083658561354474, −5.22326946124576233581010483229, −5.08155571008720359858285682504, −4.94918347073555387524553622767, −4.83092838044900378333194401575, −4.59475466893083337497267540057, −4.49406180149230968746013697756, −4.45814702933879927704555817115, −4.21485372889973320307235437118, −4.19249756990346798275667685478, −3.59273223894421960727012895838, −3.58515212804073181565946697371, −3.53362035677608930485911083418, −3.25763914098061505575982546630, −2.78558076696772335986770746362, −2.73140779158461928123933634897, −2.69693876875754153223078918788, −2.61814016416761753649833808003, −2.22464135613454089164037395193, −1.81971563627221143955425862428, −1.52810866145170798219785871179, −1.29590774620716669223231835382, −1.21797040116736838359817543015, −1.11230830272604284563183722380, 0, 0, 0, 0, 0,
1.11230830272604284563183722380, 1.21797040116736838359817543015, 1.29590774620716669223231835382, 1.52810866145170798219785871179, 1.81971563627221143955425862428, 2.22464135613454089164037395193, 2.61814016416761753649833808003, 2.69693876875754153223078918788, 2.73140779158461928123933634897, 2.78558076696772335986770746362, 3.25763914098061505575982546630, 3.53362035677608930485911083418, 3.58515212804073181565946697371, 3.59273223894421960727012895838, 4.19249756990346798275667685478, 4.21485372889973320307235437118, 4.45814702933879927704555817115, 4.49406180149230968746013697756, 4.59475466893083337497267540057, 4.83092838044900378333194401575, 4.94918347073555387524553622767, 5.08155571008720359858285682504, 5.22326946124576233581010483229, 5.33706283337834083658561354474, 5.41428776650590569393751473760
