| L(s) = 1 | − 2·3-s − 4·5-s − 4·7-s + 9-s + 6·11-s + 4·13-s + 8·15-s − 4·17-s + 3·19-s + 8·21-s − 12·23-s + 2·29-s + 2·31-s − 12·33-s + 16·35-s − 2·37-s − 8·39-s − 4·41-s + 2·43-s − 4·45-s − 6·49-s + 8·51-s + 14·53-s − 24·55-s − 6·57-s − 28·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 2.06·15-s − 0.970·17-s + 0.688·19-s + 1.74·21-s − 2.50·23-s + 0.371·29-s + 0.359·31-s − 2.08·33-s + 2.70·35-s − 0.328·37-s − 1.28·39-s − 0.624·41-s + 0.304·43-s − 0.596·45-s − 6/7·49-s + 1.12·51-s + 1.92·53-s − 3.23·55-s − 0.794·57-s − 3.64·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_d_e |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 16 T^{2} + 36 T^{3} + 16 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.e_q_bk |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 22 T^{2} + 52 T^{3} + 22 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_w_ca |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 128 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bm_aey |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_p_abo |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 40 T^{2} + 138 T^{3} + 40 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bo_fi |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.23.m_en_xs |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 33 T^{2} + 60 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ac_bh_ci |
| 31 | $S_4\times C_2$ | \( 1 - 2 T - T^{2} + 228 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ac_ab_iu |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 116 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_dt_em |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 396 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_cv_pg |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 98 T^{2} - 216 T^{3} + 98 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_du_aii |
| 47 | $S_4\times C_2$ | \( 1 + 50 T^{2} + 332 T^{3} + 50 p T^{4} + p^{3} T^{6} \) | 3.47.a_by_mu |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 145 T^{2} - 1116 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ao_fp_abqy |
| 59 | $S_4\times C_2$ | \( 1 + 28 T + 421 T^{2} + 3960 T^{3} + 421 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.bc_qf_fwi |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 108 T^{2} + 380 T^{3} + 108 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_ee_oq |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 1132 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_cx_abro |
| 71 | $S_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 1536 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.q_ht_chc |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 348 T^{2} - 3458 T^{3} + 348 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ay_nk_afda |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 964 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_hj_blc |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 285 T^{2} - 3336 T^{3} + 285 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.au_kz_aeyi |
| 89 | $S_4\times C_2$ | \( 1 + 209 T^{2} + 124 T^{3} + 209 p T^{4} + p^{3} T^{6} \) | 3.89.a_ib_eu |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 435 T^{2} + 4540 T^{3} + 435 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.w_qt_gsq |
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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
−7.75681480856065699082860459411, −7.41523289040994427804144317935, −7.10361985912247142614120348228, −6.85529817682694753822424657202, −6.67557874845470258146178642500, −6.33412778127992583116694877357, −6.29461599838534118896420630102, −5.99236472064188109001942813794, −5.85478286338167159353965135159, −5.79901475292111110312918767390, −5.13967979864362668945065198146, −4.84517832791978981321709827002, −4.72358051663273923894233348380, −4.26031125289867366225382540667, −4.16288411510771769974965845473, −3.74120306229997364821711344697, −3.70361230442267426566972955601, −3.54124165814501002870000558131, −3.31292742047757599013929947476, −2.75596417432593187263719369790, −2.49637005079455046753922161169, −1.97403182167015767209214824158, −1.61902924040344362616824778446, −1.25591092163536260685688202127, −0.884190417597860569681523378948, 0, 0, 0,
0.884190417597860569681523378948, 1.25591092163536260685688202127, 1.61902924040344362616824778446, 1.97403182167015767209214824158, 2.49637005079455046753922161169, 2.75596417432593187263719369790, 3.31292742047757599013929947476, 3.54124165814501002870000558131, 3.70361230442267426566972955601, 3.74120306229997364821711344697, 4.16288411510771769974965845473, 4.26031125289867366225382540667, 4.72358051663273923894233348380, 4.84517832791978981321709827002, 5.13967979864362668945065198146, 5.79901475292111110312918767390, 5.85478286338167159353965135159, 5.99236472064188109001942813794, 6.29461599838534118896420630102, 6.33412778127992583116694877357, 6.67557874845470258146178642500, 6.85529817682694753822424657202, 7.10361985912247142614120348228, 7.41523289040994427804144317935, 7.75681480856065699082860459411
