| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.599551372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.599551372\) |
| \(L(\frac{3}{2})\) |
|
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.ah_bf_adq |
| 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.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.ai_cb_ajg |
| 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.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.ar_hc_abug |
| 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.e_j_ei |
| 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.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.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.f_w_ako |
| 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.b_ik_ew |
| 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.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
−7.959253477440208852477444383680, −7.55776317970792062193348957069, −7.46831117319741735603407535178, −7.43460973313759157576188468306, −6.67920843118123391739510750602, −6.44421415026960976901966693936, −6.38189968620705610356306366452, −5.82742837630452335925619043608, −5.75402553305669644203269000236, −5.63952546765489272486691049621, −5.05513544279625830352625172154, −5.02991098804043194414009689911, −4.67493027157922854612594253043, −4.30859222905583820755562710161, −4.26914812625389816914390275517, −3.93216035221722942357898506867, −3.23673672930971092103283349476, −3.23147843869033772616768329954, −2.92995999126187042135093269501, −2.15788028160567918688415226779, −2.01514899171345556991368943836, −1.92415209647921171034781719101, −1.26417473221193574661965776433, −0.917300351310702402452691260785, −0.798674056178037082436618027808,
0.798674056178037082436618027808, 0.917300351310702402452691260785, 1.26417473221193574661965776433, 1.92415209647921171034781719101, 2.01514899171345556991368943836, 2.15788028160567918688415226779, 2.92995999126187042135093269501, 3.23147843869033772616768329954, 3.23673672930971092103283349476, 3.93216035221722942357898506867, 4.26914812625389816914390275517, 4.30859222905583820755562710161, 4.67493027157922854612594253043, 5.02991098804043194414009689911, 5.05513544279625830352625172154, 5.63952546765489272486691049621, 5.75402553305669644203269000236, 5.82742837630452335925619043608, 6.38189968620705610356306366452, 6.44421415026960976901966693936, 6.67920843118123391739510750602, 7.43460973313759157576188468306, 7.46831117319741735603407535178, 7.55776317970792062193348957069, 7.959253477440208852477444383680
