| 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_c |
| 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.ad_p_abe |
| 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.g_bq_fy |
| 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.ag_bh_acq |
| 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.g_bq_di |
| 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.ag_ff_atg |
| 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.g_ap_awa |
| 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.j_bn_eg |
| 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.ay_nh_aeqy |
| 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.av_nh_aety |
| 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.m_kn_cxs |
| 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_oy |
| 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
−7.01570705661680822260656346439, −6.90322675043134280697617053564, −6.83020763803188123139176981968, −6.75787394866519458692759703068, −6.25362965360793324834918312898, −6.02474073092625862023752878417, −5.86659910343364783929703059348, −5.49704443379439214687099721842, −5.33034136662350122109509034527, −5.10292980211848295314302535717, −4.79143264090657412237094586000, −4.59219205522045890101382261191, −4.50107856665445790571406803121, −3.97347868062664212677888061351, −3.71523032979897543448890864929, −3.69305404981113201081732807961, −3.39165685186836478211059322320, −3.03114092542283969911619040342, −2.79848938136265470565714156185, −2.33828572313983749006856065610, −2.27483505286464357705728411370, −2.10408031028959127198137007552, −1.54214324228190901364517818211, −1.17401364315783513978712248974, −0.986726011486835339083871197503, 0, 0, 0,
0.986726011486835339083871197503, 1.17401364315783513978712248974, 1.54214324228190901364517818211, 2.10408031028959127198137007552, 2.27483505286464357705728411370, 2.33828572313983749006856065610, 2.79848938136265470565714156185, 3.03114092542283969911619040342, 3.39165685186836478211059322320, 3.69305404981113201081732807961, 3.71523032979897543448890864929, 3.97347868062664212677888061351, 4.50107856665445790571406803121, 4.59219205522045890101382261191, 4.79143264090657412237094586000, 5.10292980211848295314302535717, 5.33034136662350122109509034527, 5.49704443379439214687099721842, 5.86659910343364783929703059348, 6.02474073092625862023752878417, 6.25362965360793324834918312898, 6.75787394866519458692759703068, 6.83020763803188123139176981968, 6.90322675043134280697617053564, 7.01570705661680822260656346439
