| L(s) = 1 | − 3·5-s − 2·7-s + 2·11-s − 4·13-s + 4·17-s − 3·19-s − 6·23-s + 6·25-s + 12·29-s + 4·31-s + 6·35-s + 2·37-s − 4·41-s − 4·43-s − 18·47-s − 5·49-s + 8·53-s − 6·55-s + 2·59-s − 22·61-s + 12·65-s − 10·67-s + 22·71-s − 16·73-s − 4·77-s + 20·79-s + 22·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 0.603·11-s − 1.10·13-s + 0.970·17-s − 0.688·19-s − 1.25·23-s + 6/5·25-s + 2.22·29-s + 0.718·31-s + 1.01·35-s + 0.328·37-s − 0.624·41-s − 0.609·43-s − 2.62·47-s − 5/7·49-s + 1.09·53-s − 0.809·55-s + 0.260·59-s − 2.81·61-s + 1.48·65-s − 1.22·67-s + 2.61·71-s − 1.87·73-s − 0.455·77-s + 2.25·79-s + 2.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 38 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_j_bm |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 31 T^{2} - 42 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_bf_abq |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 114 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.e_x_ek |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 128 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_bp_ey |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 642 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_eb_ays |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 204 T^{3} + 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_f_hw |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} + 250 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ac_t_jq |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 42 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_bx_abq |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 558 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_bp_vm |
| 47 | $S_4\times C_2$ | \( 1 + 18 T + 209 T^{2} + 1744 T^{3} + 209 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.s_ib_cpc |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 175 T^{2} - 852 T^{3} + 175 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_gt_abgu |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 125 T^{2} - 52 T^{3} + 125 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ac_ev_aca |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 311 T^{2} + 2760 T^{3} + 311 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.w_lz_ece |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 181 T^{2} + 1348 T^{3} + 181 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.k_gz_bzw |
| 71 | $S_4\times C_2$ | \( 1 - 22 T + 361 T^{2} - 3420 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aw_nx_afbo |
| 73 | $S_4\times C_2$ | \( 1 + 16 T + 155 T^{2} + 1296 T^{3} + 155 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.q_fz_bxw |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 277 T^{2} - 2696 T^{3} + 277 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.au_kr_adzs |
| 83 | $S_4\times C_2$ | \( 1 - 22 T + 333 T^{2} - 3312 T^{3} + 333 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aw_mv_aexk |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2354 T^{3} + 257 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.s_jx_dmo |
| 97 | $S_4\times C_2$ | \( 1 + 36 T + 719 T^{2} + 8666 T^{3} + 719 p T^{4} + 36 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bk_bbr_mvi |
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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
−7.69926686286384738276412989288, −6.93623474203591132015934504438, −6.81353636034155046424327182560, −6.80046586953230645762321668972, −6.52036215273647497385111400343, −6.31792262398017431299685443481, −6.10213477625072249114425595773, −5.88663419749593470735651797724, −5.26428424333200109344654096290, −5.23464656730200785736595812314, −4.95404860126049902034775167070, −4.77248809287217065376542767581, −4.51554774337324048114208868937, −4.14232189437923993358654586987, −4.00251507406423074870824041396, −3.73325423063971370598250691924, −3.48506878732526222590250602690, −3.08994032505660304591007481261, −3.02874324280652983239590635183, −2.66014164027546280490697954502, −2.34617181296717913766275802001, −2.13697334073515940797902930875, −1.37231176887609981080978566776, −1.27816052917424541138842099135, −1.06462863942519887452612833468, 0, 0, 0,
1.06462863942519887452612833468, 1.27816052917424541138842099135, 1.37231176887609981080978566776, 2.13697334073515940797902930875, 2.34617181296717913766275802001, 2.66014164027546280490697954502, 3.02874324280652983239590635183, 3.08994032505660304591007481261, 3.48506878732526222590250602690, 3.73325423063971370598250691924, 4.00251507406423074870824041396, 4.14232189437923993358654586987, 4.51554774337324048114208868937, 4.77248809287217065376542767581, 4.95404860126049902034775167070, 5.23464656730200785736595812314, 5.26428424333200109344654096290, 5.88663419749593470735651797724, 6.10213477625072249114425595773, 6.31792262398017431299685443481, 6.52036215273647497385111400343, 6.80046586953230645762321668972, 6.81353636034155046424327182560, 6.93623474203591132015934504438, 7.69926686286384738276412989288
