| L(s) = 1 | + 2·7-s − 2·9-s − 6·11-s + 3·13-s + 4·17-s − 2·19-s − 6·23-s + 4·27-s − 2·29-s − 12·31-s − 8·37-s + 2·41-s + 12·43-s + 10·47-s − 2·49-s + 6·53-s − 2·59-s + 10·61-s − 4·63-s + 12·67-s + 8·71-s − 18·73-s − 12·77-s − 28·79-s − 2·81-s + 16·83-s − 2·89-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 2/3·9-s − 1.80·11-s + 0.832·13-s + 0.970·17-s − 0.458·19-s − 1.25·23-s + 0.769·27-s − 0.371·29-s − 2.15·31-s − 1.31·37-s + 0.312·41-s + 1.82·43-s + 1.45·47-s − 2/7·49-s + 0.824·53-s − 0.260·59-s + 1.28·61-s − 0.503·63-s + 1.46·67-s + 0.949·71-s − 2.10·73-s − 1.36·77-s − 3.15·79-s − 2/9·81-s + 1.75·83-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576823227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576823227\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T^{2} - 4 T^{3} + 2 p T^{4} + p^{3} T^{6} \) | 3.3.a_c_ae |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 6 T^{2} - 8 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_g_ai |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.11.g_bt_fk |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 52 T^{3} + 8 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_i_ca |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 116 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_n_em |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 260 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cb_ka |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_br_ga |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 113 T^{2} + 664 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.m_ej_zo |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 112 T^{2} + 590 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_ei_ws |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_br_ga |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 736 T^{3} + 114 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.am_ek_abci |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 158 T^{2} - 932 T^{3} + 158 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_gc_abjw |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 143 T^{2} - 620 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fn_axw |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 133 T^{2} + 276 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_fd_kq |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ak_ff_abwe |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1528 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_in_acgu |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 178 T^{2} - 936 T^{3} + 178 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ai_gw_abka |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) | 3.73.s_mp_efk |
| 79 | $S_4\times C_2$ | \( 1 + 28 T + 453 T^{2} + 4744 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.bc_rl_ham |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 289 T^{2} - 2496 T^{3} + 289 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aq_ld_adsa |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 223 T^{2} + 396 T^{3} + 223 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_ip_pg |
| 97 | $S_4\times C_2$ | \( 1 + 26 T + 343 T^{2} + 3452 T^{3} + 343 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ba_nf_fcu |
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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
−7.33423943207495916499338384158, −7.27308422713902884925491368925, −6.80073973687176131220015888831, −6.62353607695155856059981860275, −6.05524162048514507318319460531, −5.94300873971735319840932821626, −5.78537461286839190851890895065, −5.56264359410273855670000475808, −5.33834073522465036191965074199, −5.18217905131547094084143841125, −4.92164297160461954164606092112, −4.41074107268811680826057808208, −4.33113943259900117115475393936, −3.78142885254990020942159275974, −3.76985012843789175954644669887, −3.66909513755982831007188551749, −2.92082227696003968631602358148, −2.75866755916981861086652347420, −2.73675590259825544373563152721, −2.12809377527793856960556157325, −1.99579500745269488570612329828, −1.58272995434998929451597458323, −1.23778228727632873548565789437, −0.67039629935881879907764764494, −0.26404127968188573385603359855,
0.26404127968188573385603359855, 0.67039629935881879907764764494, 1.23778228727632873548565789437, 1.58272995434998929451597458323, 1.99579500745269488570612329828, 2.12809377527793856960556157325, 2.73675590259825544373563152721, 2.75866755916981861086652347420, 2.92082227696003968631602358148, 3.66909513755982831007188551749, 3.76985012843789175954644669887, 3.78142885254990020942159275974, 4.33113943259900117115475393936, 4.41074107268811680826057808208, 4.92164297160461954164606092112, 5.18217905131547094084143841125, 5.33834073522465036191965074199, 5.56264359410273855670000475808, 5.78537461286839190851890895065, 5.94300873971735319840932821626, 6.05524162048514507318319460531, 6.62353607695155856059981860275, 6.80073973687176131220015888831, 7.27308422713902884925491368925, 7.33423943207495916499338384158
