| L(s) = 1 | + 2·3-s − 3·5-s − 6·7-s − 9-s + 4·11-s − 3·13-s − 6·15-s − 2·17-s + 4·19-s − 12·21-s − 6·23-s + 6·25-s − 4·27-s − 2·29-s + 8·33-s + 18·35-s − 2·37-s − 6·39-s + 2·41-s + 6·43-s + 3·45-s − 14·47-s + 3·49-s − 4·51-s − 2·53-s − 12·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 2.26·7-s − 1/3·9-s + 1.20·11-s − 0.832·13-s − 1.54·15-s − 0.485·17-s + 0.917·19-s − 2.61·21-s − 1.25·23-s + 6/5·25-s − 0.769·27-s − 0.371·29-s + 1.39·33-s + 3.04·35-s − 0.328·37-s − 0.960·39-s + 0.312·41-s + 0.914·43-s + 0.447·45-s − 2.04·47-s + 3/7·49-s − 0.560·51-s − 0.274·53-s − 1.61·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_f_ai |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.7.g_bh_do |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 84 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ae_bh_adg |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_p_abk |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 148 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ae_cf_afs |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 128 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_bp_ey |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} - 68 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_bj_acq |
| 31 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 76 T^{3} + 53 p T^{4} + p^{3} T^{6} \) | 3.31.a_cb_cy |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} - 52 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_t_aca |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 87 T^{2} - 60 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_dj_aci |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 101 T^{2} - 520 T^{3} + 101 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ag_dx_aua |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 153 T^{2} + 1164 T^{3} + 153 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.o_fx_bsu |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 139 T^{2} + 204 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.c_fj_hw |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1308 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.am_gv_abyi |
| 61 | $D_{6}$ | \( 1 + 6 T + 51 T^{2} + 20 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_bz_u |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} + 412 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_cr_pw |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 221 T^{2} + 1684 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_in_cmu |
| 73 | $S_4\times C_2$ | \( 1 + 22 T + 367 T^{2} + 3508 T^{3} + 367 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.w_od_fey |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 336 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ai_ev_amy |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 101 T^{2} + 868 T^{3} + 101 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.c_dx_bhk |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 55 T^{2} + 1404 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_cd_cca |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 239 T^{2} + 1908 T^{3} + 239 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.w_jf_cvk |
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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
−7.905483033831485212648370988335, −7.52755684372677323594634929774, −7.41775642731824672714046606531, −7.04468380185335789712256043017, −6.88626241755345564706537674578, −6.63549323703630744232340745394, −6.48282837946609588385076841983, −6.03631711420829239404401889503, −5.97279870194189594886933812090, −5.79176543208294284453697041985, −5.16829602243189682556822856850, −5.05449184134693655427229261382, −4.74681741671350292955430272202, −4.32358983558303772187625652703, −4.00964944939865496765332889009, −3.98349895576892032263441421187, −3.50368771460645650775426985588, −3.35505218234393207587257982514, −3.26990678967364565104347129881, −2.78667305850368228415995521355, −2.55603868433244882877832320772, −2.52788649271123976856536750529, −1.81042896300726631976392781590, −1.38463470183708103876051508359, −1.08588073865503884739429056818, 0, 0, 0,
1.08588073865503884739429056818, 1.38463470183708103876051508359, 1.81042896300726631976392781590, 2.52788649271123976856536750529, 2.55603868433244882877832320772, 2.78667305850368228415995521355, 3.26990678967364565104347129881, 3.35505218234393207587257982514, 3.50368771460645650775426985588, 3.98349895576892032263441421187, 4.00964944939865496765332889009, 4.32358983558303772187625652703, 4.74681741671350292955430272202, 5.05449184134693655427229261382, 5.16829602243189682556822856850, 5.79176543208294284453697041985, 5.97279870194189594886933812090, 6.03631711420829239404401889503, 6.48282837946609588385076841983, 6.63549323703630744232340745394, 6.88626241755345564706537674578, 7.04468380185335789712256043017, 7.41775642731824672714046606531, 7.52755684372677323594634929774, 7.905483033831485212648370988335
