| L(s) = 1 | + 2-s − 8·7-s − 2·11-s − 4·13-s − 8·14-s − 16-s + 4·17-s − 4·19-s − 2·22-s + 6·23-s − 4·26-s + 2·29-s + 3·31-s + 32-s + 4·34-s − 10·37-s − 4·38-s + 8·41-s − 10·43-s + 6·46-s − 10·47-s + 27·49-s + 2·58-s + 8·59-s − 6·61-s + 3·62-s − 2·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3.02·7-s − 0.603·11-s − 1.10·13-s − 2.13·14-s − 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.426·22-s + 1.25·23-s − 0.784·26-s + 0.371·29-s + 0.538·31-s + 0.176·32-s + 0.685·34-s − 1.64·37-s − 0.648·38-s + 1.24·41-s − 1.52·43-s + 0.884·46-s − 1.45·47-s + 27/7·49-s + 0.262·58-s + 1.04·59-s − 0.768·61-s + 0.381·62-s − 1/4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_b_ab |
| 7 | $S_4\times C_2$ | \( 1 + 8 T + 37 T^{2} + 118 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.i_bl_eo |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 20 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_n_u |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 98 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.e_bn_du |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 132 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_afc |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 56 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_z_ce |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 272 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_cf_akm |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 230 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ac_v_aiw |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 139 T^{2} + 758 T^{3} + 139 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_fj_bde |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 608 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ai_et_axk |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 141 T^{2} + 836 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.k_fl_bge |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.k_dl_qq |
| 53 | $S_4\times C_2$ | \( 1 + 135 T^{2} - 36 T^{3} + 135 p T^{4} + p^{3} T^{6} \) | 3.53.a_ff_abk |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 187 T^{2} - 938 T^{3} + 187 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ai_hf_abkc |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.61.g_hn_bcm |
| 67 | $S_4\times C_2$ | \( 1 + 18 T + 273 T^{2} + 2330 T^{3} + 273 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.s_kn_dlq |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 1162 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_fr_bss |
| 73 | $S_4\times C_2$ | \( 1 + 18 T + 279 T^{2} + 2522 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.s_kt_dta |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 333 T^{2} - 3124 T^{3} + 333 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.au_mv_aeqe |
| 83 | $S_4\times C_2$ | \( 1 - 22 T + 325 T^{2} - 3160 T^{3} + 325 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aw_mn_aero |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.i_f_be |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 123 T^{2} + 64 T^{3} + 123 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.i_et_cm |
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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
−7.34461406294318280794107432959, −6.87428527845573863017337086635, −6.82097504266487837799537874792, −6.64192377420199934969948885708, −6.63491855319621580392200679546, −6.08438431190071659457424460988, −5.98554766316627719078278177930, −5.88180989733868710214182075310, −5.43320902250591339258467182789, −5.29115444807525774618656681065, −4.85792984740046202021297321787, −4.69500520995715511114339683155, −4.66509513950840020920005048917, −4.28834707543148489095900183539, −3.83123315366894664058616670791, −3.65337530601741352712503282861, −3.34590824963253091979956212094, −3.13685605144588961950395001449, −3.04898611974851492795808742462, −2.69968392051932364185223378862, −2.57121363586191369805889463996, −1.97683051332487313314013389890, −1.87482602700970157324992308456, −1.10527275274349620774310162147, −0.988147011255202477562530668569, 0, 0, 0,
0.988147011255202477562530668569, 1.10527275274349620774310162147, 1.87482602700970157324992308456, 1.97683051332487313314013389890, 2.57121363586191369805889463996, 2.69968392051932364185223378862, 3.04898611974851492795808742462, 3.13685605144588961950395001449, 3.34590824963253091979956212094, 3.65337530601741352712503282861, 3.83123315366894664058616670791, 4.28834707543148489095900183539, 4.66509513950840020920005048917, 4.69500520995715511114339683155, 4.85792984740046202021297321787, 5.29115444807525774618656681065, 5.43320902250591339258467182789, 5.88180989733868710214182075310, 5.98554766316627719078278177930, 6.08438431190071659457424460988, 6.63491855319621580392200679546, 6.64192377420199934969948885708, 6.82097504266487837799537874792, 6.87428527845573863017337086635, 7.34461406294318280794107432959
