| L(s) = 1 | + 2-s − 2·4-s − 2·7-s − 2·8-s − 6·11-s + 2·13-s − 2·14-s + 16-s − 6·22-s + 2·23-s + 2·26-s + 4·28-s − 8·29-s + 3·31-s − 32-s − 8·41-s + 10·43-s + 12·44-s + 2·46-s − 6·47-s − 15·49-s − 4·52-s + 4·56-s − 8·58-s − 14·59-s − 2·61-s + 3·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.755·7-s − 0.707·8-s − 1.80·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.27·22-s + 0.417·23-s + 0.392·26-s + 0.755·28-s − 1.48·29-s + 0.538·31-s − 0.176·32-s − 1.24·41-s + 1.52·43-s + 1.80·44-s + 0.294·46-s − 0.875·47-s − 2.14·49-s − 0.554·52-s + 0.534·56-s − 1.05·58-s − 1.82·59-s − 0.256·61-s + 0.381·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_d_ad |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 26 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_t_ba |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.11.g_bt_fk |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 17 T^{2} - 54 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_r_acc |
| 17 | $S_4\times C_2$ | \( 1 + 23 T^{2} + 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \) | 3.17.a_x_ca |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.19.a_cf_a |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 88 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ac_cn_adk |
| 29 | $S_4\times C_2$ | \( 1 + 8 T + 71 T^{2} + 334 T^{3} + 71 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.i_ct_mw |
| 37 | $S_4\times C_2$ | \( 1 - 7 T^{2} + 358 T^{3} - 7 p T^{4} + p^{3} T^{6} \) | 3.37.a_ah_nu |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 91 T^{2} + 528 T^{3} + 91 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.i_dn_ui |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 852 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_ev_abgu |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 496 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_eb_tc |
| 53 | $S_4\times C_2$ | \( 1 + 119 T^{2} + 76 T^{3} + 119 p T^{4} + p^{3} T^{6} \) | 3.53.a_ep_cy |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 149 T^{2} + 982 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.o_ft_blu |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 147 T^{2} + 140 T^{3} + 147 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_fr_fk |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 203 T^{2} - 1842 T^{3} + 203 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aq_hv_acsw |
| 71 | $S_4\times C_2$ | \( 1 + 20 T + 297 T^{2} + 2706 T^{3} + 297 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.u_ll_eac |
| 73 | $S_4\times C_2$ | \( 1 + 149 T^{2} - 86 T^{3} + 149 p T^{4} + p^{3} T^{6} \) | 3.73.a_ft_adi |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 145 T^{2} - 884 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ai_fp_abia |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 117 T^{2} - 336 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ac_en_amy |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 83 T^{2} + 2 T^{3} + 83 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_df_c |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) | 3.97.abk_bbv_amxc |
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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
−7.46204491122111712583361452490, −7.03384945539502868544046238666, −6.94900759820036747110681987552, −6.61445485503462271045324375994, −6.35963975527518870178408288859, −6.06390088885855882096725084878, −6.04908150527819694849604987795, −5.58442063299519439737090647975, −5.33985071932240847474394849970, −5.33232268331931568463788583482, −4.82819335139974406541011627274, −4.78452315927123161398707979617, −4.67893067772266809225876693530, −4.17308871497055973180974324057, −4.10113061551044184170404355610, −3.66977219552438767209883282014, −3.31261516881152094632871440041, −3.31015447868465147053041711440, −3.13449898224920744790392401722, −2.53614409617627261213368552688, −2.48137204834009977009648708536, −1.97258035879966970335370864856, −1.77986441779354911505611110712, −1.09353592348674401071897324503, −1.06499528321804395605277008259, 0, 0, 0,
1.06499528321804395605277008259, 1.09353592348674401071897324503, 1.77986441779354911505611110712, 1.97258035879966970335370864856, 2.48137204834009977009648708536, 2.53614409617627261213368552688, 3.13449898224920744790392401722, 3.31015447868465147053041711440, 3.31261516881152094632871440041, 3.66977219552438767209883282014, 4.10113061551044184170404355610, 4.17308871497055973180974324057, 4.67893067772266809225876693530, 4.78452315927123161398707979617, 4.82819335139974406541011627274, 5.33232268331931568463788583482, 5.33985071932240847474394849970, 5.58442063299519439737090647975, 6.04908150527819694849604987795, 6.06390088885855882096725084878, 6.35963975527518870178408288859, 6.61445485503462271045324375994, 6.94900759820036747110681987552, 7.03384945539502868544046238666, 7.46204491122111712583361452490
