| L(s) = 1 | + 3-s + 7·5-s − 3·7-s − 6·9-s + 5·11-s + 13-s + 7·15-s − 8·17-s + 3·19-s − 3·21-s + 20·25-s − 8·27-s + 29-s − 31-s + 5·33-s − 21·35-s − 13·37-s + 39-s − 21·41-s − 16·43-s − 42·45-s + 3·47-s − 15·49-s − 8·51-s + 8·53-s + 35·55-s + 3·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.13·5-s − 1.13·7-s − 2·9-s + 1.50·11-s + 0.277·13-s + 1.80·15-s − 1.94·17-s + 0.688·19-s − 0.654·21-s + 4·25-s − 1.53·27-s + 0.185·29-s − 0.179·31-s + 0.870·33-s − 3.54·35-s − 2.13·37-s + 0.160·39-s − 3.27·41-s − 2.43·43-s − 6.26·45-s + 0.437·47-s − 2.14·49-s − 1.12·51-s + 1.09·53-s + 4.71·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 151 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - T + 7 T^{2} - 5 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_h_af |
| 5 | $A_4\times C_2$ | \( 1 - 7 T + 29 T^{2} - 77 T^{3} + 29 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ah_bd_acz |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) | 3.7.d_y_br |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 32 T^{2} - 97 T^{3} + 32 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.af_bg_adt |
| 13 | $A_4\times C_2$ | \( 1 - T + 23 T^{2} - 3 p T^{3} + 23 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ab_x_abn |
| 17 | $A_4\times C_2$ | \( 1 + 8 T + 56 T^{2} + 229 T^{3} + 56 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_ce_iv |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 11 T^{2} + 25 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_l_z |
| 23 | $A_4\times C_2$ | \( 1 + 48 T^{2} - 7 T^{3} + 48 p T^{4} + p^{3} T^{6} \) | 3.23.a_bw_ah |
| 29 | $C_6$ | \( 1 - T + 15 T^{2} - 99 T^{3} + 15 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ab_p_adv |
| 31 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 19 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.31.b_cl_t |
| 37 | $A_4\times C_2$ | \( 1 + 13 T + 151 T^{2} + 991 T^{3} + 151 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.n_fv_bmd |
| 41 | $A_4\times C_2$ | \( 1 + 21 T + 242 T^{2} + 1813 T^{3} + 242 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.v_ji_crt |
| 43 | $A_4\times C_2$ | \( 1 + 16 T + 170 T^{2} + 1179 T^{3} + 170 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.q_go_btj |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 32 T^{2} + 277 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ad_bg_kr |
| 53 | $A_4\times C_2$ | \( 1 - 8 T + 136 T^{2} - 651 T^{3} + 136 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_fg_azb |
| 59 | $A_4\times C_2$ | \( 1 - T + 77 T^{2} - 299 T^{3} + 77 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ab_cz_aln |
| 61 | $A_4\times C_2$ | \( 1 - T + 125 T^{2} - 135 T^{3} + 125 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ab_ev_aff |
| 67 | $A_4\times C_2$ | \( 1 - T + 31 T^{2} - 175 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ab_bf_agt |
| 71 | $A_4\times C_2$ | \( 1 + 14 T + 164 T^{2} + 1099 T^{3} + 164 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.o_gi_bqh |
| 73 | $A_4\times C_2$ | \( 1 + T + 154 T^{2} - 23 T^{3} + 154 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_fy_ax |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 149 T^{2} + 181 T^{3} + 149 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_ft_gz |
| 83 | $A_4\times C_2$ | \( 1 - 3 T + 224 T^{2} - 5 p T^{3} + 224 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ad_iq_apz |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) | 3.89.bk_bax_may |
| 97 | $A_4\times C_2$ | \( 1 + 228 T^{2} + 189 T^{3} + 228 p T^{4} + p^{3} T^{6} \) | 3.97.a_iu_hh |
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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
−6.92492853014520703423478666680, −6.70788798725343152859897221114, −6.62073299849410154462778860440, −6.45772895786453263615996833363, −6.08355155243497505925879039608, −6.07491666484155793317749509385, −5.96181399628910004901209792858, −5.55642792544031375231893613085, −5.20920759011548214154050156170, −5.12967657088166316102893489992, −5.10473599314150649204724963035, −4.74864841300514961937685555684, −4.30606965360106095244956699751, −3.80891581398256420682719636631, −3.73309480085691679817594645990, −3.52571834712969159246947208715, −3.13109473678983628875002417919, −2.94315198592942340593890581356, −2.87294306834380713182675593505, −2.33478127424002618491637628648, −2.09886008284187791165326641330, −2.05112701941344336463609473667, −1.55928336557307726433918176265, −1.36724046078515150810370170054, −1.31522504331637075760202052620, 0, 0, 0,
1.31522504331637075760202052620, 1.36724046078515150810370170054, 1.55928336557307726433918176265, 2.05112701941344336463609473667, 2.09886008284187791165326641330, 2.33478127424002618491637628648, 2.87294306834380713182675593505, 2.94315198592942340593890581356, 3.13109473678983628875002417919, 3.52571834712969159246947208715, 3.73309480085691679817594645990, 3.80891581398256420682719636631, 4.30606965360106095244956699751, 4.74864841300514961937685555684, 5.10473599314150649204724963035, 5.12967657088166316102893489992, 5.20920759011548214154050156170, 5.55642792544031375231893613085, 5.96181399628910004901209792858, 6.07491666484155793317749509385, 6.08355155243497505925879039608, 6.45772895786453263615996833363, 6.62073299849410154462778860440, 6.70788798725343152859897221114, 6.92492853014520703423478666680
