| L(s) = 1 | − 3·7-s − 3·11-s − 6·13-s + 6·17-s − 6·23-s + 6·29-s − 15·37-s − 6·41-s − 3·43-s + 6·47-s + 6·49-s + 9·53-s − 24·59-s − 6·61-s + 9·67-s − 21·71-s − 12·73-s + 9·77-s − 15·79-s + 12·83-s + 18·91-s + 30·97-s + 24·101-s − 18·103-s − 15·107-s − 18·109-s + 15·113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s − 1.66·13-s + 1.45·17-s − 1.25·23-s + 1.11·29-s − 2.46·37-s − 0.937·41-s − 0.457·43-s + 0.875·47-s + 6/7·49-s + 1.23·53-s − 3.12·59-s − 0.768·61-s + 1.09·67-s − 2.49·71-s − 1.40·73-s + 1.02·77-s − 1.68·79-s + 1.31·83-s + 1.88·91-s + 3.04·97-s + 2.38·101-s − 1.77·103-s − 1.45·107-s − 1.72·109-s + 1.41·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $D_{6}$ | \( 1 - 2 T^{3} + p^{3} T^{6} \) | 3.5.a_a_ac |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 30 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_p_be |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 132 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bk_fc |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 154 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ag_bq_afy |
| 19 | $S_4\times C_2$ | \( 1 + 33 T^{2} - 8 T^{3} + 33 p T^{4} + p^{3} T^{6} \) | 3.19.a_bh_ai |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 68 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_bh_cq |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 86 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_bq_adi |
| 31 | $S_4\times C_2$ | \( 1 + 69 T^{2} - 8 T^{3} + 69 p T^{4} + p^{3} T^{6} \) | 3.31.a_cr_ai |
| 37 | $S_4\times C_2$ | \( 1 + 15 T + 162 T^{2} + 1107 T^{3} + 162 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.p_gg_bqp |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.41.g_ff_tg |
| 43 | $D_{6}$ | \( 1 + 3 T + 9 T^{2} - 142 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.d_j_afm |
| 47 | $S_4\times C_2$ | \( 1 - 6 T - 15 T^{2} + 572 T^{3} - 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_ap_wa |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 39 T^{2} - 110 T^{3} + 39 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_bn_aeg |
| 59 | $S_4\times C_2$ | \( 1 + 24 T + 345 T^{2} + 3144 T^{3} + 345 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.y_nh_eqy |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 120 T^{2} + 344 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_eq_ng |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 171 T^{2} - 1218 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aj_gp_abuw |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 345 T^{2} + 3222 T^{3} + 345 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.v_nh_ety |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 246 T^{2} + 1748 T^{3} + 246 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.m_jm_cpg |
| 79 | $S_4\times C_2$ | \( 1 + 15 T + 291 T^{2} + 2374 T^{3} + 291 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.p_lf_dni |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 273 T^{2} - 1968 T^{3} + 273 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.am_kn_acxs |
| 89 | $S_4\times C_2$ | \( 1 + 162 T^{2} - 388 T^{3} + 162 p T^{4} + p^{3} T^{6} \) | 3.89.a_gg_aoy |
| 97 | $D_{6}$ | \( 1 - 30 T + 399 T^{2} - 3940 T^{3} + 399 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.abe_pj_afvo |
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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
−7.30963077239234002357814343961, −7.21186820361115461805083030298, −6.61386861219166735562167230589, −6.51897492476161439373747194842, −6.22187572460583309282778633961, −5.93916933791741309516327729965, −5.91943229456816115604867862333, −5.58125274244432424931328855003, −5.30122490920361243262361979788, −5.07861091797419117593264692593, −4.76603348213284437864375063595, −4.73068051281876478113122169750, −4.53230783244496127732396778473, −3.85818461468851423168299188056, −3.78637170697199936060338703412, −3.73677054326128038474278059590, −3.11232090960945531944508095496, −2.98423432250501212523969814818, −2.96824651441374267761849170577, −2.58315868151779488379138575631, −2.17727102506332337434823693686, −1.99489337187056953725389050473, −1.58552563041772523140189025820, −1.25027861497327474001887598039, −0.873912889994800420346329979746, 0, 0, 0,
0.873912889994800420346329979746, 1.25027861497327474001887598039, 1.58552563041772523140189025820, 1.99489337187056953725389050473, 2.17727102506332337434823693686, 2.58315868151779488379138575631, 2.96824651441374267761849170577, 2.98423432250501212523969814818, 3.11232090960945531944508095496, 3.73677054326128038474278059590, 3.78637170697199936060338703412, 3.85818461468851423168299188056, 4.53230783244496127732396778473, 4.73068051281876478113122169750, 4.76603348213284437864375063595, 5.07861091797419117593264692593, 5.30122490920361243262361979788, 5.58125274244432424931328855003, 5.91943229456816115604867862333, 5.93916933791741309516327729965, 6.22187572460583309282778633961, 6.51897492476161439373747194842, 6.61386861219166735562167230589, 7.21186820361115461805083030298, 7.30963077239234002357814343961
