| L(s) = 1 | + 2·5-s + 4·7-s − 6·13-s + 2·17-s − 10·19-s + 12·23-s − 3·25-s + 6·29-s + 14·31-s + 8·35-s − 3·37-s + 20·41-s + 2·43-s + 8·47-s + 3·49-s + 4·59-s − 18·61-s − 12·65-s + 4·67-s + 16·71-s − 6·73-s + 14·79-s + 4·85-s + 10·89-s − 24·91-s − 20·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.66·13-s + 0.485·17-s − 2.29·19-s + 2.50·23-s − 3/5·25-s + 1.11·29-s + 2.51·31-s + 1.35·35-s − 0.493·37-s + 3.12·41-s + 0.304·43-s + 1.16·47-s + 3/7·49-s + 0.520·59-s − 2.30·61-s − 1.48·65-s + 0.488·67-s + 1.89·71-s − 0.702·73-s + 1.57·79-s + 0.433·85-s + 1.05·89-s − 2.51·91-s − 2.05·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.607704086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.607704086\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 24 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_h_ay |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 40 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_n_abo |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.11.a_bh_a |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.13.g_bz_gi |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 120 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_t_aeq |
| 19 | $S_4\times C_2$ | \( 1 + 10 T + 77 T^{2} + 372 T^{3} + 77 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.k_cz_oi |
| 23 | $S_4\times C_2$ | \( 1 - 12 T + 89 T^{2} - 452 T^{3} + 89 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.am_dl_ark |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 71 T^{2} - 352 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_ct_ano |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 97 T^{2} - 524 T^{3} + 97 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ao_dt_aue |
| 41 | $S_4\times C_2$ | \( 1 - 20 T + 235 T^{2} - 1800 T^{3} + 235 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.au_jb_acrg |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 117 T^{2} - 164 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_en_agi |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 592 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_ev_awu |
| 53 | $S_4\times C_2$ | \( 1 + 95 T^{2} - 128 T^{3} + 95 p T^{4} + p^{3} T^{6} \) | 3.53.a_dr_aey |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 149 T^{2} - 356 T^{3} + 149 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ae_ft_ans |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 227 T^{2} + 1900 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.s_it_cvc |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 145 T^{2} - 616 T^{3} + 145 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ae_fp_axs |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 261 T^{2} - 2304 T^{3} + 261 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_kb_adkq |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 164 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_dj_gi |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 209 T^{2} - 1740 T^{3} + 209 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ao_ib_acoy |
| 83 | $S_4\times C_2$ | \( 1 + 137 T^{2} - 416 T^{3} + 137 p T^{4} + p^{3} T^{6} \) | 3.83.a_fh_aqa |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 291 T^{2} - 1776 T^{3} + 291 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ak_lf_acqi |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 239 T^{2} + 916 T^{3} + 239 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_jf_bjg |
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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
−7.908590150448821360102425053190, −7.55313766488929788569763639443, −7.37763836677900692332549540421, −7.26036440606107311980532374124, −6.68849004691746522000455517968, −6.47064863037016067796641675675, −6.44005156300462442172439924597, −6.00766059595221197757649591256, −5.72362879681482289725319967018, −5.59435314346150838388079970455, −4.99348151854920319545625062002, −4.82192361078317692314264403995, −4.76907836455326258342977024796, −4.57316482060796946062107412522, −4.14522479582286275677232000466, −4.01663778737228849229312981596, −3.26244347043203943909231692483, −3.06965619798971730935291609016, −2.72734219278629536722890530368, −2.29935369163514749324350978642, −2.10674236593404839371059000626, −2.03859012821566471634895281243, −1.28885735964439102228054436796, −0.810650343140465164743951011758, −0.68004654339980161998734801695,
0.68004654339980161998734801695, 0.810650343140465164743951011758, 1.28885735964439102228054436796, 2.03859012821566471634895281243, 2.10674236593404839371059000626, 2.29935369163514749324350978642, 2.72734219278629536722890530368, 3.06965619798971730935291609016, 3.26244347043203943909231692483, 4.01663778737228849229312981596, 4.14522479582286275677232000466, 4.57316482060796946062107412522, 4.76907836455326258342977024796, 4.82192361078317692314264403995, 4.99348151854920319545625062002, 5.59435314346150838388079970455, 5.72362879681482289725319967018, 6.00766059595221197757649591256, 6.44005156300462442172439924597, 6.47064863037016067796641675675, 6.68849004691746522000455517968, 7.26036440606107311980532374124, 7.37763836677900692332549540421, 7.55313766488929788569763639443, 7.908590150448821360102425053190
