| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(84.18723343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(84.18723343\) |
| \(L(\frac{3}{2})\) |
|
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.ae_m_abc_cd_aec |
| 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.ac_x_abs_ny_aqe |
| 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.ae_bu_afe_bil_adec |
| 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.ag_cm_aec_clr_achc |
| 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.ag_dj_aug_ifs_abkom |
| 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.am_ed_abse_nfy_adalk |
| 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.as_he_acvs_znj_ahkzq |
| 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.ai_jg_acpi_blmd_aiprg |
| 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.c_kr_to_bvoo_crou |
| 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.ak_jw_adfi_bqel_alkmg |
| 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.ai_id_adro_bibm_aqwhk |
| 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
−4.61290459196152708463568108107, −4.43651776910465323165503638201, −4.40548933699646528307352793461, −4.31960381067805832117646393261, −4.31224204510337163619006014664, −4.07020662580350112082193078542, −3.61482390338805504454729189795, −3.60283436426330752055759599949, −3.46570836443418083716668959372, −3.24203962183610341734728730455, −2.87611055352162256480396468265, −2.78710467386997000120806640909, −2.63712544612822535941808946890, −2.61551997333010992654600081429, −2.59802015547729433316790380890, −2.32279983149029754413364944079, −2.03512782353930172438065076411, −1.94239544962094570968932147477, −1.73686685906974836148517358823, −1.62682249471496282441888944854, −1.39500721012441549276771296797, −1.15482233636226196929798596924, −0.71148670818245232967518408785, −0.66516431469268541983840734608, −0.54791363524363720304891799546,
0.54791363524363720304891799546, 0.66516431469268541983840734608, 0.71148670818245232967518408785, 1.15482233636226196929798596924, 1.39500721012441549276771296797, 1.62682249471496282441888944854, 1.73686685906974836148517358823, 1.94239544962094570968932147477, 2.03512782353930172438065076411, 2.32279983149029754413364944079, 2.59802015547729433316790380890, 2.61551997333010992654600081429, 2.63712544612822535941808946890, 2.78710467386997000120806640909, 2.87611055352162256480396468265, 3.24203962183610341734728730455, 3.46570836443418083716668959372, 3.60283436426330752055759599949, 3.61482390338805504454729189795, 4.07020662580350112082193078542, 4.31224204510337163619006014664, 4.31960381067805832117646393261, 4.40548933699646528307352793461, 4.43651776910465323165503638201, 4.61290459196152708463568108107
