| L(s) = 1 | − 3-s + 3·5-s − 7-s − 2·9-s + 5·13-s − 3·15-s + 5·17-s + 3·19-s + 21-s − 23-s + 6·25-s + 3·27-s + 17·29-s + 2·31-s − 3·35-s + 8·37-s − 5·39-s + 6·41-s − 10·43-s − 6·45-s + 4·47-s − 8·49-s − 5·51-s + 5·53-s − 3·57-s + 15·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.377·7-s − 2/3·9-s + 1.38·13-s − 0.774·15-s + 1.21·17-s + 0.688·19-s + 0.218·21-s − 0.208·23-s + 6/5·25-s + 0.577·27-s + 3.15·29-s + 0.359·31-s − 0.507·35-s + 1.31·37-s − 0.800·39-s + 0.937·41-s − 1.52·43-s − 0.894·45-s + 0.583·47-s − 8/7·49-s − 0.700·51-s + 0.686·53-s − 0.397·57-s + 1.95·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.117580238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.117580238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 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.b_d_c |
| 7 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} - 2 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.7.b_j_ac |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.11.a_bh_a |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 41 T^{2} - 126 T^{3} + 41 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.af_bp_aew |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 47 T^{2} - 166 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.af_bv_agk |
| 23 | $S_4\times C_2$ | \( 1 + T + 37 T^{2} - 18 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.23.b_bl_as |
| 29 | $S_4\times C_2$ | \( 1 - 17 T + 171 T^{2} - 1110 T^{3} + 171 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ar_gp_abqs |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 45 T^{2} + 4 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ac_bt_e |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_eh_awm |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ct_aqu |
| 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.k_fh_bfw |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 504 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ae_cj_atk |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 117 T^{2} - 582 T^{3} + 117 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.af_en_awk |
| 59 | $S_4\times C_2$ | \( 1 - 15 T + 149 T^{2} - 986 T^{3} + 149 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ap_ft_ably |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ai_gx_abjc |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 23 T^{2} + 650 T^{3} + 23 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ad_x_za |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 21 T^{2} - 456 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_v_aro |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 119 T^{2} - 10 p T^{3} + 119 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ad_ep_abcc |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 213 T^{2} - 284 T^{3} + 213 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_if_aky |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 233 T^{2} + 1628 T^{3} + 233 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.k_iz_ckq |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_ih_abia |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 187 T^{2} + 1072 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_hf_bpg |
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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
−9.339608443846956485332514463948, −8.815038998151942066650079151664, −8.681384638026606490842714315350, −8.389629143354915464958881966870, −8.021247988585065163288621818349, −8.009450292800877235090905566131, −7.43257497693855931358395546361, −6.88725203173550604459093765087, −6.77035964981026072585441356101, −6.46868095677231596224256272830, −6.07836165716150442674164725842, −6.07433500239069738366994744695, −5.62868764048575192137593915729, −5.24576155296133299179287079208, −5.14529422628463535960614763253, −4.79315188530252213002848303014, −4.09783949361175612796706099188, −3.95142913392650859996777779527, −3.46187234887498844093351198676, −2.90155047877099722007122839638, −2.62516524298711256247397858528, −2.55099828486234539448364061704, −1.49818681625352366274665120402, −1.20344395864318515258100406463, −0.75797977384676029689490623979,
0.75797977384676029689490623979, 1.20344395864318515258100406463, 1.49818681625352366274665120402, 2.55099828486234539448364061704, 2.62516524298711256247397858528, 2.90155047877099722007122839638, 3.46187234887498844093351198676, 3.95142913392650859996777779527, 4.09783949361175612796706099188, 4.79315188530252213002848303014, 5.14529422628463535960614763253, 5.24576155296133299179287079208, 5.62868764048575192137593915729, 6.07433500239069738366994744695, 6.07836165716150442674164725842, 6.46868095677231596224256272830, 6.77035964981026072585441356101, 6.88725203173550604459093765087, 7.43257497693855931358395546361, 8.009450292800877235090905566131, 8.021247988585065163288621818349, 8.389629143354915464958881966870, 8.681384638026606490842714315350, 8.815038998151942066650079151664, 9.339608443846956485332514463948
