| L(s) = 1 | − 3·3-s − 3·5-s − 6·7-s + 6·9-s − 6·11-s + 9·15-s − 6·17-s + 18·21-s + 6·23-s − 6·25-s − 10·27-s + 15·29-s − 3·31-s + 18·33-s + 18·35-s + 12·37-s + 6·43-s − 18·45-s + 6·49-s + 18·51-s − 18·53-s + 18·55-s − 9·59-s − 15·61-s − 36·63-s + 6·67-s − 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 2.26·7-s + 2·9-s − 1.80·11-s + 2.32·15-s − 1.45·17-s + 3.92·21-s + 1.25·23-s − 6/5·25-s − 1.92·27-s + 2.78·29-s − 0.538·31-s + 3.13·33-s + 3.04·35-s + 1.97·37-s + 0.914·43-s − 2.68·45-s + 6/7·49-s + 2.52·51-s − 2.47·53-s + 2.42·55-s − 1.17·59-s − 1.92·61-s − 4.53·63-s + 0.733·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + p T + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_d_b |
| 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_p_bb |
| 7 | $A_4\times C_2$ | \( 1 + 6 T + 30 T^{2} + 87 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.g_be_dj |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 133 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bq_fd |
| 13 | $A_4\times C_2$ | \( 1 + 18 T^{2} - 17 T^{3} + 18 p T^{4} + p^{3} T^{6} \) | 3.13.a_s_ar |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 185 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.g_bk_hd |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 140 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_bt_afk |
| 29 | $A_4\times C_2$ | \( 1 - 15 T + 159 T^{2} - 981 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ap_gd_ablt |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 133 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.d_cr_fd |
| 37 | $A_4\times C_2$ | \( 1 - 12 T + 150 T^{2} - 907 T^{3} + 150 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.am_fu_abix |
| 41 | $A_4\times C_2$ | \( 1 + 48 T^{2} - 125 T^{3} + 48 p T^{4} + p^{3} T^{6} \) | 3.41.a_bw_aev |
| 43 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 427 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ag_dy_aql |
| 47 | $A_4\times C_2$ | \( 1 + 102 T^{2} + 89 T^{3} + 102 p T^{4} + p^{3} T^{6} \) | 3.47.a_dy_dl |
| 53 | $A_4\times C_2$ | \( 1 + 18 T + 186 T^{2} + 1395 T^{3} + 186 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.s_he_cbr |
| 59 | $A_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 541 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.j_eq_uv |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 210 T^{2} + 1651 T^{3} + 210 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.p_ic_cln |
| 67 | $A_4\times C_2$ | \( 1 - 6 T - 39 T^{2} + 1060 T^{3} - 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_abn_bou |
| 71 | $A_4\times C_2$ | \( 1 - 6 T - 27 T^{2} + 1012 T^{3} - 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ag_abb_bmy |
| 73 | $A_4\times C_2$ | \( 1 + 75 T^{2} + 576 T^{3} + 75 p T^{4} + p^{3} T^{6} \) | 3.73.a_cx_we |
| 79 | $A_4\times C_2$ | \( 1 - 21 T + 321 T^{2} - 3391 T^{3} + 321 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.av_mj_afal |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 294 T^{2} + 2045 T^{3} + 294 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.m_li_dar |
| 89 | $A_4\times C_2$ | \( 1 - 15 T + 321 T^{2} - 2727 T^{3} + 321 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_mj_aeax |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 402 T^{2} - 4093 T^{3} + 402 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.av_pm_agbl |
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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
−8.165825996533120381004533625112, −7.66771165971377169882993394206, −7.65322276815200665295603517871, −7.41913072303918237946131865531, −7.17209935479662508354608376019, −6.65706961362532692084811101056, −6.46629723713251312982080041924, −6.34348919287883573642103803432, −6.26123061165351441514478160313, −6.08514423385117476452260731951, −5.54608864784582521472175667019, −5.29771711631864362201101297493, −4.98436393272257365946456769140, −4.65708652413809832592916483116, −4.60146940775065241963721786535, −4.35857479346081744042476748639, −3.80289808970907939468973341004, −3.62945736263725863025645023026, −3.40631347002566547068377528484, −2.75317411359468363673634007767, −2.69912018669026324762169800580, −2.61121391180836378220404653151, −1.86424175406958215066630692768, −1.23937457787683583822921354658, −0.840673988290036197681387588239, 0, 0, 0,
0.840673988290036197681387588239, 1.23937457787683583822921354658, 1.86424175406958215066630692768, 2.61121391180836378220404653151, 2.69912018669026324762169800580, 2.75317411359468363673634007767, 3.40631347002566547068377528484, 3.62945736263725863025645023026, 3.80289808970907939468973341004, 4.35857479346081744042476748639, 4.60146940775065241963721786535, 4.65708652413809832592916483116, 4.98436393272257365946456769140, 5.29771711631864362201101297493, 5.54608864784582521472175667019, 6.08514423385117476452260731951, 6.26123061165351441514478160313, 6.34348919287883573642103803432, 6.46629723713251312982080041924, 6.65706961362532692084811101056, 7.17209935479662508354608376019, 7.41913072303918237946131865531, 7.65322276815200665295603517871, 7.66771165971377169882993394206, 8.165825996533120381004533625112
