| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 6·5-s − 9·6-s + 3·7-s − 10·8-s + 6·9-s + 18·10-s + 18·12-s − 6·13-s − 9·14-s − 18·15-s + 15·16-s − 6·17-s − 18·18-s − 36·20-s + 9·21-s + 6·23-s − 30·24-s + 12·25-s + 18·26-s + 10·27-s + 18·28-s − 6·29-s + 54·30-s − 15·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 2.68·5-s − 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s + 5.69·10-s + 5.19·12-s − 1.66·13-s − 2.40·14-s − 4.64·15-s + 15/4·16-s − 1.45·17-s − 4.24·18-s − 8.04·20-s + 1.96·21-s + 1.25·23-s − 6.12·24-s + 12/5·25-s + 3.53·26-s + 1.92·27-s + 3.40·28-s − 1.11·29-s + 9.85·30-s − 2.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | | \( 1 \) | |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 61 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.g_y_cj |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 15 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_d_p |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 37 T^{3} + 12 p T^{4} + p^{3} T^{6} \) | 3.11.a_m_bl |
| 13 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 155 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bq_fz |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.g_ci_hz |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 121 T^{3} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_a_er |
| 29 | $A_4\times C_2$ | \( 1 + 6 T + 96 T^{2} + 349 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_ds_nl |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 165 T^{2} + 1039 T^{3} + 165 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.p_gj_bnz |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 66 T^{2} - 239 T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_co_ajf |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 685 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.j_ez_baj |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 523 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_dv_aud |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 775 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.j_ez_bdv |
| 53 | $A_4\times C_2$ | \( 1 + 27 T + 390 T^{2} + 3491 T^{3} + 390 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.bb_pa_feh |
| 59 | $A_4\times C_2$ | \( 1 + 18 T + 222 T^{2} + 2133 T^{3} + 222 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.s_io_deb |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 165 T^{2} - 383 T^{3} + 165 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_gj_aot |
| 67 | $A_4\times C_2$ | \( 1 + 12 T - 24 T^{2} - 1041 T^{3} - 24 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.m_ay_abob |
| 71 | $A_4\times C_2$ | \( 1 + 21 T + 303 T^{2} + 2819 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.v_lr_eel |
| 73 | $A_4\times C_2$ | \( 1 - 15 T + 273 T^{2} - 2247 T^{3} + 273 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ap_kn_adil |
| 79 | $A_4\times C_2$ | \( 1 + 15 T + 249 T^{2} + 2189 T^{3} + 249 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.p_jp_dgf |
| 83 | $A_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 549 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.d_if_vd |
| 89 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 45 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.d_et_bt |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1861 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.m_gm_ctp |
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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
−8.487891795271399064389099630923, −7.966909408000182404942354026344, −7.88117770456729180213250836696, −7.83629873970740420070325534626, −7.55757077431523067542781561777, −7.52628565688876195874712402192, −7.25041380751551384716072791126, −6.83065106060343318885704359227, −6.71873350100179112019338495573, −6.47438348487866476885529950586, −5.87008986697618749579546758723, −5.44511009356410320974645685518, −5.16702082388057134630603910200, −4.74485012026844724971329136532, −4.42414007622867928505867185650, −4.36880452256356266630146223308, −3.87699313684063147141512762888, −3.55366846371205034770337556494, −3.42336755975861852588881822535, −2.81761896189157385117483533914, −2.71533465635953895299994306865, −2.43118401458852033039421428224, −1.67155374630645298126687353118, −1.53181406821201147370422732928, −1.49963772018727642169634911075, 0, 0, 0,
1.49963772018727642169634911075, 1.53181406821201147370422732928, 1.67155374630645298126687353118, 2.43118401458852033039421428224, 2.71533465635953895299994306865, 2.81761896189157385117483533914, 3.42336755975861852588881822535, 3.55366846371205034770337556494, 3.87699313684063147141512762888, 4.36880452256356266630146223308, 4.42414007622867928505867185650, 4.74485012026844724971329136532, 5.16702082388057134630603910200, 5.44511009356410320974645685518, 5.87008986697618749579546758723, 6.47438348487866476885529950586, 6.71873350100179112019338495573, 6.83065106060343318885704359227, 7.25041380751551384716072791126, 7.52628565688876195874712402192, 7.55757077431523067542781561777, 7.83629873970740420070325534626, 7.88117770456729180213250836696, 7.966909408000182404942354026344, 8.487891795271399064389099630923
