| L(s) = 1 | − 2·3-s − 5-s − 7·7-s − 9-s + 3·13-s + 2·15-s − 4·17-s − 8·19-s + 14·21-s − 9·23-s − 9·25-s + 7·27-s − 9·29-s − 17·31-s + 7·35-s − 3·37-s − 6·39-s − 16·41-s + 4·43-s + 45-s − 11·47-s + 18·49-s + 8·51-s − 3·53-s + 16·57-s + 2·59-s + 15·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 2.64·7-s − 1/3·9-s + 0.832·13-s + 0.516·15-s − 0.970·17-s − 1.83·19-s + 3.05·21-s − 1.87·23-s − 9/5·25-s + 1.34·27-s − 1.67·29-s − 3.05·31-s + 1.18·35-s − 0.493·37-s − 0.960·39-s − 2.49·41-s + 0.609·43-s + 0.149·45-s − 1.60·47-s + 18/7·49-s + 1.12·51-s − 0.412·53-s + 2.11·57-s + 0.260·59-s + 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_f |
| 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.b_k_m |
| 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_abb |
| 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.e_bf_eq |
| 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.j_do_qm |
| 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.j_eg_uq |
| 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.q_hl_cah |
| 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.l_fb_bnq |
| 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.d_ch_xm |
| 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.ac_cb_aim |
| 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.af_hh_abbm |
| 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.aj_gr_acju |
| 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.aq_kh_aedc |
| 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
−9.990392712974872045625074044138, −9.811486788187792446629135767706, −9.398585634354791434476865898227, −9.222582370381474991729878684161, −8.726122921778138655536023254043, −8.470537108293830054377683717722, −8.440211798153525048876671744691, −7.75412954347892405880640985076, −7.63528508692796755069757850058, −6.97946680295344347948617800115, −6.82483640366394813721241584182, −6.60160560001282227479028780829, −6.29213539355547884989312976495, −5.80238789685692823543845840333, −5.78513746162881769781831084105, −5.76524329481190551015197117579, −4.83299823781801531441695991198, −4.82752699320630119816144663728, −3.93349317524837682261595292904, −3.63242059709245727016449970833, −3.60563061350904574298556263715, −3.49892796007126454839896413076, −2.51269391462931742691739591812, −2.00990782041008292729706732962, −1.84176972901464465199741885513, 0, 0, 0,
1.84176972901464465199741885513, 2.00990782041008292729706732962, 2.51269391462931742691739591812, 3.49892796007126454839896413076, 3.60563061350904574298556263715, 3.63242059709245727016449970833, 3.93349317524837682261595292904, 4.82752699320630119816144663728, 4.83299823781801531441695991198, 5.76524329481190551015197117579, 5.78513746162881769781831084105, 5.80238789685692823543845840333, 6.29213539355547884989312976495, 6.60160560001282227479028780829, 6.82483640366394813721241584182, 6.97946680295344347948617800115, 7.63528508692796755069757850058, 7.75412954347892405880640985076, 8.440211798153525048876671744691, 8.470537108293830054377683717722, 8.726122921778138655536023254043, 9.222582370381474991729878684161, 9.398585634354791434476865898227, 9.811486788187792446629135767706, 9.990392712974872045625074044138
