| L(s) = 1 | + 3-s − 2·9-s − 2·11-s − 5·13-s − 3·17-s − 11·19-s − 3·27-s + 5·29-s − 3·31-s − 2·33-s − 4·37-s − 5·39-s + 6·41-s + 10·43-s − 25·47-s − 5·49-s − 3·51-s − 19·53-s − 11·57-s − 7·59-s + 3·61-s − 10·67-s + 25·71-s − 9·73-s + 2·79-s − 4·81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.603·11-s − 1.38·13-s − 0.727·17-s − 2.52·19-s − 0.577·27-s + 0.928·29-s − 0.538·31-s − 0.348·33-s − 0.657·37-s − 0.800·39-s + 0.937·41-s + 1.52·43-s − 3.64·47-s − 5/7·49-s − 0.420·51-s − 2.60·53-s − 1.45·57-s − 0.911·59-s + 0.384·61-s − 1.22·67-s + 2.96·71-s − 1.05·73-s + 0.225·79-s − 4/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 2 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_d_ac |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 8 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.7.a_f_ai |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 60 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_n_ci |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 35 T^{2} + 126 T^{3} + 35 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.f_bj_ew |
| 19 | $S_4\times C_2$ | \( 1 + 11 T + 85 T^{2} + 434 T^{3} + 85 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.l_dh_qs |
| 23 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 8 T^{3} + 53 p T^{4} + p^{3} T^{6} \) | 3.23.a_cb_ai |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 63 T^{2} - 294 T^{3} + 63 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_cl_ali |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 35 T^{2} + 158 T^{3} + 35 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.d_bj_gc |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 67 T^{2} + 328 T^{3} + 67 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.e_cp_mq |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.41.ag_ff_atg |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 137 T^{2} - 828 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_fh_abfw |
| 47 | $S_4\times C_2$ | \( 1 + 25 T + 317 T^{2} + 2606 T^{3} + 317 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.z_mf_dwg |
| 53 | $S_4\times C_2$ | \( 1 + 19 T + 247 T^{2} + 2010 T^{3} + 247 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.t_jn_czi |
| 59 | $S_4\times C_2$ | \( 1 + 7 T + 69 T^{2} + 234 T^{3} + 69 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.h_cr_ja |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 143 T^{2} - 218 T^{3} + 143 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_fn_aik |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 185 T^{2} + 1308 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.k_hd_byi |
| 71 | $S_4\times C_2$ | \( 1 - 25 T + 371 T^{2} - 3578 T^{3} + 371 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.az_oh_afhq |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 203 T^{2} + 1318 T^{3} + 203 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.j_hv_bys |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 177 T^{2} - 92 T^{3} + 177 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_gv_ado |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 2592 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.q_jp_dvs |
| 89 | $S_4\times C_2$ | \( 1 - 15 T + 239 T^{2} - 2674 T^{3} + 239 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_jf_adyw |
| 97 | $S_4\times C_2$ | \( 1 - 13 T + 195 T^{2} - 1630 T^{3} + 195 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.an_hn_acks |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38850367231586561100063056246, −7.24853992397151804035954860383, −6.92945047983595493801891540730, −6.56100939102451876973044018493, −6.33508679264076676884143814447, −6.31307250549864942259402874978, −6.29731209292396828932026220588, −5.71573326265023360922686941219, −5.41403959744838301284728935933, −5.27570209226588252492521866382, −4.79995825768013865113956139083, −4.77251723860020793086661602597, −4.65996165970713228135256172342, −4.15076263005340830144114995472, −4.14442131801277568116637120528, −3.72762728424397999963212257718, −3.21824927327330266865932510211, −3.08715849138481122589035371295, −3.03628224829540817617087904220, −2.43323947868659603321431717790, −2.30302714546367605728625233597, −2.21138151492614445645559690706, −1.66959537067622283648872596168, −1.54023151797889733941253284734, −0.902665140938255736494760134006, 0, 0, 0,
0.902665140938255736494760134006, 1.54023151797889733941253284734, 1.66959537067622283648872596168, 2.21138151492614445645559690706, 2.30302714546367605728625233597, 2.43323947868659603321431717790, 3.03628224829540817617087904220, 3.08715849138481122589035371295, 3.21824927327330266865932510211, 3.72762728424397999963212257718, 4.14442131801277568116637120528, 4.15076263005340830144114995472, 4.65996165970713228135256172342, 4.77251723860020793086661602597, 4.79995825768013865113956139083, 5.27570209226588252492521866382, 5.41403959744838301284728935933, 5.71573326265023360922686941219, 6.29731209292396828932026220588, 6.31307250549864942259402874978, 6.33508679264076676884143814447, 6.56100939102451876973044018493, 6.92945047983595493801891540730, 7.24853992397151804035954860383, 7.38850367231586561100063056246
