| L(s) = 1 | + 3·3-s − 3·5-s − 4·7-s + 6·9-s + 2·11-s + 4·13-s − 9·15-s − 2·17-s − 4·19-s − 12·21-s + 2·23-s + 6·25-s + 10·27-s + 3·31-s + 6·33-s + 12·35-s + 6·37-s + 12·39-s − 8·41-s − 18·43-s − 18·45-s + 4·47-s + 49-s − 6·51-s + 26·53-s − 6·55-s − 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s + 0.603·11-s + 1.10·13-s − 2.32·15-s − 0.485·17-s − 0.917·19-s − 2.61·21-s + 0.417·23-s + 6/5·25-s + 1.92·27-s + 0.538·31-s + 1.04·33-s + 2.02·35-s + 0.986·37-s + 1.92·39-s − 1.24·41-s − 2.74·43-s − 2.68·45-s + 0.583·47-s + 1/7·49-s − 0.840·51-s + 3.57·53-s − 0.809·55-s − 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.105782250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.105782250\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} + 38 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_p_bm |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 20 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_n_au |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 86 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_bh_adi |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 56 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_br_ce |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 56 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_z_ce |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 61 T^{2} - 80 T^{3} + 61 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ac_cj_adc |
| 29 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 34 T^{3} + 39 p T^{4} + p^{3} T^{6} \) | 3.29.a_bn_bi |
| 37 | $S_4\times C_2$ | \( 1 - 6 T - 3 T^{2} + 278 T^{3} - 3 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_ad_ks |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 608 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.i_et_xk |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 165 T^{2} + 1124 T^{3} + 165 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.s_gj_brg |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 116 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ae_v_em |
| 53 | $S_4\times C_2$ | \( 1 - 26 T + 347 T^{2} - 3000 T^{3} + 347 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aba_nj_aelk |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 141 T^{2} - 586 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_fl_awo |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) | 3.61.as_lf_adou |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 155 T^{2} + 1378 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.k_fz_cba |
| 71 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 954 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.71.a_j_bks |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 369 T^{2} - 3518 T^{3} + 369 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aw_of_affi |
| 79 | $S_4\times C_2$ | \( 1 + 153 T^{2} + 164 T^{3} + 153 p T^{4} + p^{3} T^{6} \) | 3.79.a_fx_gi |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 1032 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.g_fx_bns |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 195 T^{2} - 1882 T^{3} + 195 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ak_hn_acuk |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} + 200 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ae_bj_hs |
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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
−7.03828718849902434381234136435, −6.71999209539449819791135887969, −6.69333408640981534234942318573, −6.50746329164085266777259353477, −6.19706689748894889805216710145, −5.83408846097366136239833628923, −5.66736023395041101560531102209, −5.15765418795859068609034204473, −5.01410942417554372059712424592, −4.78386991215907635836972561612, −4.28252549157512214805938041308, −4.21584154010321863438948333421, −3.95440006729741291792051448579, −3.58607168136397941856338405122, −3.56659977952359790508136304686, −3.53929560517869347805469075174, −2.85130922659840765975946986259, −2.80899573263421474706684359032, −2.74860287181644136617916102452, −1.98397817881356103391245809802, −1.83308172709289703058280887016, −1.80123511186182102379915962571, −0.854233977983507907235061486468, −0.74186666919694817558768396944, −0.48876487698826236554111861339,
0.48876487698826236554111861339, 0.74186666919694817558768396944, 0.854233977983507907235061486468, 1.80123511186182102379915962571, 1.83308172709289703058280887016, 1.98397817881356103391245809802, 2.74860287181644136617916102452, 2.80899573263421474706684359032, 2.85130922659840765975946986259, 3.53929560517869347805469075174, 3.56659977952359790508136304686, 3.58607168136397941856338405122, 3.95440006729741291792051448579, 4.21584154010321863438948333421, 4.28252549157512214805938041308, 4.78386991215907635836972561612, 5.01410942417554372059712424592, 5.15765418795859068609034204473, 5.66736023395041101560531102209, 5.83408846097366136239833628923, 6.19706689748894889805216710145, 6.50746329164085266777259353477, 6.69333408640981534234942318573, 6.71999209539449819791135887969, 7.03828718849902434381234136435
