| L(s) = 1 | + 5-s − 7·7-s + 3·13-s + 4·17-s − 8·19-s + 9·23-s − 9·25-s + 9·29-s − 17·31-s − 7·35-s − 3·37-s + 16·41-s + 4·43-s + 11·47-s + 18·49-s + 3·53-s − 2·59-s + 15·61-s + 3·65-s + 5·67-s − 5·71-s − 6·73-s + 79-s − 9·83-s + 4·85-s − 16·89-s − 21·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2.64·7-s + 0.832·13-s + 0.970·17-s − 1.83·19-s + 1.87·23-s − 9/5·25-s + 1.67·29-s − 3.05·31-s − 1.18·35-s − 0.493·37-s + 2.49·41-s + 0.609·43-s + 1.60·47-s + 18/7·49-s + 0.412·53-s − 0.260·59-s + 1.92·61-s + 0.372·65-s + 0.610·67-s − 0.593·71-s − 0.702·73-s + 0.112·79-s − 0.987·83-s + 0.433·85-s − 1.69·89-s − 2.20·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254078307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254078307\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 12 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ab_k_am |
| 7 | $S_4\times C_2$ | \( 1 + p T + 31 T^{2} + 94 T^{3} + 31 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.7.h_bf_dq |
| 11 | $S_4\times C_2$ | \( 1 - 3 T^{2} + 27 T^{3} - 3 p T^{4} + p^{3} T^{6} \) | 3.11.a_ad_bb |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 16 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_g_aq |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 120 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bf_aeq |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 240 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cb_jg |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 428 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.aj_do_aqm |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 110 T^{2} - 536 T^{3} + 110 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.aj_eg_auq |
| 31 | $S_4\times C_2$ | \( 1 + 17 T + 184 T^{2} + 1202 T^{3} + 184 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.r_hc_bug |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 193 T^{2} - 1359 T^{3} + 193 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aq_hl_acah |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 9 T^{2} - 112 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_j_aei |
| 47 | $S_4\times C_2$ | \( 1 - 11 T + 131 T^{2} - 1030 T^{3} + 131 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.al_fb_abnq |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 59 T^{2} - 610 T^{3} + 59 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ad_ch_axm |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_cb_im |
| 61 | $S_4\times C_2$ | \( 1 - 15 T + 212 T^{2} - 1778 T^{3} + 212 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ap_ie_acqk |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 22 T^{2} + 274 T^{3} + 22 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.af_w_ko |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 714 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.f_hh_bbm |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 195 T^{2} + 839 T^{3} + 195 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_hn_bgh |
| 79 | $S_4\times C_2$ | \( 1 - T + 218 T^{2} - 126 T^{3} + 218 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ab_ik_aew |
| 83 | $S_4\times C_2$ | \( 1 + 9 T + 173 T^{2} + 1606 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.j_gr_cju |
| 89 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 2784 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.q_kh_edc |
| 97 | $S_4\times C_2$ | \( 1 + 47 T^{2} + 256 T^{3} + 47 p T^{4} + p^{3} T^{6} \) | 3.97.a_bv_jw |
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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
−7.13018083936678832492157122340, −6.92488077992164382953433199878, −6.76491666301517644242975341110, −6.62311515496208300952676044165, −6.09781385806852191060410178972, −5.97652949228643584187065706982, −5.97532231839144114266298116304, −5.60569374026689417425079001836, −5.48218045317538745469090931585, −5.24799421894193152942774562427, −4.57565561904431135551298515089, −4.41520413152730357002021061013, −4.30958533832296906537475705485, −3.68202607384570357004343979385, −3.66512249656618300614717943352, −3.60864743274534499251785312848, −3.12797907241128258340066524837, −2.81991539231159485517161345796, −2.65630618686658983239483735307, −2.15419359668828913617163626201, −2.11001974763906929613940279168, −1.52230106796931354858179433712, −1.05428002283421695563091071332, −0.70803596171218856070955163931, −0.24554127306581287892304654382,
0.24554127306581287892304654382, 0.70803596171218856070955163931, 1.05428002283421695563091071332, 1.52230106796931354858179433712, 2.11001974763906929613940279168, 2.15419359668828913617163626201, 2.65630618686658983239483735307, 2.81991539231159485517161345796, 3.12797907241128258340066524837, 3.60864743274534499251785312848, 3.66512249656618300614717943352, 3.68202607384570357004343979385, 4.30958533832296906537475705485, 4.41520413152730357002021061013, 4.57565561904431135551298515089, 5.24799421894193152942774562427, 5.48218045317538745469090931585, 5.60569374026689417425079001836, 5.97532231839144114266298116304, 5.97652949228643584187065706982, 6.09781385806852191060410178972, 6.62311515496208300952676044165, 6.76491666301517644242975341110, 6.92488077992164382953433199878, 7.13018083936678832492157122340
