| L(s) = 1 | − 3·3-s − 3·5-s − 3·7-s − 2·8-s + 6·9-s + 9·15-s − 12·19-s + 9·21-s + 6·24-s + 6·25-s − 10·27-s + 6·29-s − 3·31-s + 9·35-s − 6·37-s + 6·40-s + 9·43-s − 18·45-s + 12·47-s − 9·49-s + 6·56-s + 36·57-s + 6·59-s − 3·61-s − 18·63-s − 4·64-s − 9·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.13·7-s − 0.707·8-s + 2·9-s + 2.32·15-s − 2.75·19-s + 1.96·21-s + 1.22·24-s + 6/5·25-s − 1.92·27-s + 1.11·29-s − 0.538·31-s + 1.52·35-s − 0.986·37-s + 0.948·40-s + 1.37·43-s − 2.68·45-s + 1.75·47-s − 9/7·49-s + 0.801·56-s + 4.76·57-s + 0.781·59-s − 0.384·61-s − 2.26·63-s − 1/2·64-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $D_{6}$ | \( 1 + p T^{3} + p^{3} T^{6} \) | 3.2.a_a_c |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_s_bn |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 16 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.11.a_j_aq |
| 17 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 98 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.17.a_j_adu |
| 19 | $S_4\times C_2$ | \( 1 + 12 T + 93 T^{2} + 460 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.m_dp_rs |
| 23 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 96 T^{3} + 21 p T^{4} + p^{3} T^{6} \) | 3.23.a_v_ds |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 334 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_dj_amw |
| 31 | $S_4\times C_2$ | \( 1 + 3 T - 177 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.d_a_agv |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 452 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_cx_rk |
| 41 | $S_4\times C_2$ | \( 1 + 99 T^{2} - 26 T^{3} + 99 p T^{4} + p^{3} T^{6} \) | 3.41.a_dv_aba |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 763 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_fo_abdj |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 135 T^{2} - 922 T^{3} + 135 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.am_ff_abjm |
| 53 | $S_4\times C_2$ | \( 1 + 135 T^{2} + 16 T^{3} + 135 p T^{4} + p^{3} T^{6} \) | 3.53.a_ff_q |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 165 T^{2} - 694 T^{3} + 165 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_gj_abas |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 162 T^{2} + 299 T^{3} + 162 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.d_gg_ln |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 855 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.j_eq_bgx |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1022 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.am_gj_abni |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 312 T^{2} + 2977 T^{3} + 312 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.v_ma_ekn |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 144 T^{2} + 111 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_fo_eh |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 309 T^{2} - 2820 T^{3} + 309 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_lx_aeem |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 555 T^{2} + 6222 T^{3} + 555 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.be_vj_jfi |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 282 T^{2} + 2321 T^{3} + 282 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_kw_dlh |
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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
−8.235517357906576023342250645686, −8.055579378049151206181560362436, −7.85348716224467545220265014594, −7.38742178270855074768189938776, −6.89710669684316323689596444264, −6.85062055422141728850339563735, −6.83780467948919043886386498010, −6.65912748048949837892824869182, −6.01594598271490642684742414682, −6.00887681376906974272063831289, −5.82933584927721864462464589875, −5.35301096090062668599669603065, −5.34494141174692305963573933505, −4.57799276765654370971995329772, −4.50258634074334725665557886816, −4.40098531911199241818831950131, −4.03786187754529046062804614201, −3.78115364673898168714355200209, −3.42912065230080098521962631921, −2.96503395626382682106119639552, −2.84025871202651566995468063624, −2.27684914938783985233972618313, −2.00771881181323644453411162729, −1.22814428007368003786265667405, −1.03320299377090700416540949133, 0, 0, 0,
1.03320299377090700416540949133, 1.22814428007368003786265667405, 2.00771881181323644453411162729, 2.27684914938783985233972618313, 2.84025871202651566995468063624, 2.96503395626382682106119639552, 3.42912065230080098521962631921, 3.78115364673898168714355200209, 4.03786187754529046062804614201, 4.40098531911199241818831950131, 4.50258634074334725665557886816, 4.57799276765654370971995329772, 5.34494141174692305963573933505, 5.35301096090062668599669603065, 5.82933584927721864462464589875, 6.00887681376906974272063831289, 6.01594598271490642684742414682, 6.65912748048949837892824869182, 6.83780467948919043886386498010, 6.85062055422141728850339563735, 6.89710669684316323689596444264, 7.38742178270855074768189938776, 7.85348716224467545220265014594, 8.055579378049151206181560362436, 8.235517357906576023342250645686
