| L(s) = 1 | − 2·5-s − 5·7-s − 6·9-s − 4·11-s − 6·13-s − 5·17-s − 6·19-s + 18·23-s − 8·25-s + 6·27-s − 14·29-s + 16·31-s + 10·35-s + 2·37-s − 10·41-s − 20·43-s + 12·45-s + 26·47-s + 15·49-s + 8·55-s + 8·61-s + 30·63-s + 12·65-s + 12·67-s + 10·71-s − 14·73-s + 20·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.88·7-s − 2·9-s − 1.20·11-s − 1.66·13-s − 1.21·17-s − 1.37·19-s + 3.75·23-s − 8/5·25-s + 1.15·27-s − 2.59·29-s + 2.87·31-s + 1.69·35-s + 0.328·37-s − 1.56·41-s − 3.04·43-s + 1.78·45-s + 3.79·47-s + 15/7·49-s + 1.07·55-s + 1.02·61-s + 3.77·63-s + 1.48·65-s + 1.46·67-s + 1.18·71-s − 1.63·73-s + 2.27·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033604321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033604321\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 17 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 2 p T^{2} - 2 p T^{3} + 4 p T^{4} - 34 T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \) | 5.3.a_g_ag_m_abi |
| 5 | $C_2 \wr S_5$ | \( 1 + 2 T + 12 T^{2} + 32 T^{3} + 104 T^{4} + 192 T^{5} + 104 p T^{6} + 32 p^{2} T^{7} + 12 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.5.c_m_bg_ea_hk |
| 11 | $C_2 \wr S_5$ | \( 1 + 4 T + 3 p T^{2} + 100 T^{3} + 584 T^{4} + 1432 T^{5} + 584 p T^{6} + 100 p^{2} T^{7} + 3 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.e_bh_dw_wm_cdc |
| 13 | $C_2 \wr S_5$ | \( 1 + 6 T + 63 T^{2} + 288 T^{3} + 1624 T^{4} + 5468 T^{5} + 1624 p T^{6} + 288 p^{2} T^{7} + 63 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.g_cl_lc_ckm_ici |
| 19 | $C_2 \wr S_5$ | \( 1 + 6 T + 59 T^{2} + 252 T^{3} + 1714 T^{4} + 6092 T^{5} + 1714 p T^{6} + 252 p^{2} T^{7} + 59 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.g_ch_js_cny_jai |
| 23 | $C_2 \wr S_5$ | \( 1 - 18 T + 187 T^{2} - 1436 T^{3} + 9014 T^{4} - 47476 T^{5} + 9014 p T^{6} - 1436 p^{2} T^{7} + 187 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.as_hf_acdg_nis_acsga |
| 29 | $C_2 \wr S_5$ | \( 1 + 14 T + 155 T^{2} + 1224 T^{3} + 8844 T^{4} + 50676 T^{5} + 8844 p T^{6} + 1224 p^{2} T^{7} + 155 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.o_fz_bvc_nce_cwzc |
| 31 | $C_2 \wr S_5$ | \( 1 - 16 T + 168 T^{2} - 1306 T^{3} + 8698 T^{4} - 50332 T^{5} + 8698 p T^{6} - 1306 p^{2} T^{7} + 168 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.aq_gm_abyg_mwo_acwlw |
| 37 | $C_2 \wr S_5$ | \( 1 - 2 T + 97 T^{2} + 100 T^{3} + 4118 T^{4} + 11260 T^{5} + 4118 p T^{6} + 100 p^{2} T^{7} + 97 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.ac_dt_dw_gck_qrc |
| 41 | $C_2 \wr S_5$ | \( 1 + 10 T + 196 T^{2} + 1504 T^{3} + 15980 T^{4} + 89574 T^{5} + 15980 p T^{6} + 1504 p^{2} T^{7} + 196 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.k_ho_cfw_xqq_fcne |
| 43 | $C_2 \wr S_5$ | \( 1 + 20 T + 222 T^{2} + 1940 T^{3} + 17068 T^{4} + 126380 T^{5} + 17068 p T^{6} + 1940 p^{2} T^{7} + 222 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.u_io_cwq_zgm_heyu |
| 47 | $C_2 \wr S_5$ | \( 1 - 26 T + 381 T^{2} - 4040 T^{3} + 34256 T^{4} - 249340 T^{5} + 34256 p T^{6} - 4040 p^{2} T^{7} + 381 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.aba_or_afzk_byro_aoewa |
| 53 | $C_2 \wr S_5$ | \( 1 + 120 T^{2} - 132 T^{3} + 9960 T^{4} - 1290 T^{5} + 9960 p T^{6} - 132 p^{2} T^{7} + 120 p^{3} T^{8} + p^{5} T^{10} \) | 5.53.a_eq_afc_otc_abxq |
| 59 | $C_2 \wr S_5$ | \( 1 + 95 T^{2} - 96 T^{3} + 8818 T^{4} - 7744 T^{5} + 8818 p T^{6} - 96 p^{2} T^{7} + 95 p^{3} T^{8} + p^{5} T^{10} \) | 5.59.a_dr_ads_nbe_allw |
| 61 | $C_2 \wr S_5$ | \( 1 - 8 T + 232 T^{2} - 1134 T^{3} + 22076 T^{4} - 1300 p T^{5} + 22076 p T^{6} - 1134 p^{2} T^{7} + 232 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.ai_iy_abrq_bgrc_aenia |
| 67 | $C_2 \wr S_5$ | \( 1 - 12 T + 196 T^{2} - 1684 T^{3} + 19018 T^{4} - 124852 T^{5} + 19018 p T^{6} - 1684 p^{2} T^{7} + 196 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.am_ho_acmu_bcdm_ahcsa |
| 71 | $C_2 \wr S_5$ | \( 1 - 10 T + 287 T^{2} - 2468 T^{3} + 37074 T^{4} - 252148 T^{5} + 37074 p T^{6} - 2468 p^{2} T^{7} + 287 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.ak_lb_adqy_ccvy_aojaa |
| 73 | $C_2 \wr S_5$ | \( 1 + 14 T + 284 T^{2} + 2612 T^{3} + 35508 T^{4} + 259746 T^{5} + 35508 p T^{6} + 2612 p^{2} T^{7} + 284 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.o_ky_dwm_cans_ougg |
| 79 | $C_2 \wr S_5$ | \( 1 - 34 T + 845 T^{2} - 13632 T^{3} + 178016 T^{4} - 1740172 T^{5} + 178016 p T^{6} - 13632 p^{2} T^{7} + 845 p^{3} T^{8} - 34 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.abi_bgn_auei_kdiu_advafs |
| 83 | $C_2 \wr S_5$ | \( 1 - 20 T + 309 T^{2} - 4572 T^{3} + 636 p T^{4} - 482880 T^{5} + 636 p^{2} T^{6} - 4572 p^{2} T^{7} + 309 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.au_lx_agtw_daci_abbmii |
| 89 | $C_2 \wr S_5$ | \( 1 + 4 T + 199 T^{2} + 1388 T^{3} + 28716 T^{4} + 135528 T^{5} + 28716 p T^{6} + 1388 p^{2} T^{7} + 199 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.e_hr_cbk_bqmm_hsmq |
| 97 | $C_2 \wr S_5$ | \( 1 + 12 T + 268 T^{2} + 1970 T^{3} + 39168 T^{4} + 270438 T^{5} + 39168 p T^{6} + 1970 p^{2} T^{7} + 268 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.m_ki_cxu_cfym_pkbm |
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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
−4.98508383298449197682382907258, −4.83536350457802589470833175917, −4.79285062487045666522811837181, −4.58230140505840740072620066707, −4.57350507496847517440348404183, −4.12906106194391031114378340207, −3.95882351194394840981186771505, −3.74391280507409439234229234939, −3.68517672044222529327403898490, −3.56868947277915367643466219661, −3.24711950465633811094158661376, −3.08809053841196655082215872324, −3.02744270139888931916966146088, −2.82412179562553741933908870071, −2.65100109612148479965209598621, −2.34206228897425742150960122620, −2.29092901338734130014451993169, −2.24609233559777976148596128071, −2.00494591765451778286788690222, −1.68402542734146027172521200670, −1.20481512937232326646169834092, −0.69992889452417944263626881270, −0.65627448520506595961241429799, −0.39073425807401728711869271077, −0.29111587629453793691566558876,
0.29111587629453793691566558876, 0.39073425807401728711869271077, 0.65627448520506595961241429799, 0.69992889452417944263626881270, 1.20481512937232326646169834092, 1.68402542734146027172521200670, 2.00494591765451778286788690222, 2.24609233559777976148596128071, 2.29092901338734130014451993169, 2.34206228897425742150960122620, 2.65100109612148479965209598621, 2.82412179562553741933908870071, 3.02744270139888931916966146088, 3.08809053841196655082215872324, 3.24711950465633811094158661376, 3.56868947277915367643466219661, 3.68517672044222529327403898490, 3.74391280507409439234229234939, 3.95882351194394840981186771505, 4.12906106194391031114378340207, 4.57350507496847517440348404183, 4.58230140505840740072620066707, 4.79285062487045666522811837181, 4.83536350457802589470833175917, 4.98508383298449197682382907258
