| L(s) = 1 | + 3·3-s − 6·5-s + 3·7-s + 6·11-s − 18·15-s − 6·19-s + 9·21-s − 3·23-s + 12·25-s − 10·27-s + 3·29-s + 12·31-s + 18·33-s − 18·35-s + 18·37-s + 9·41-s + 15·43-s − 6·47-s + 6·49-s − 9·53-s − 36·55-s − 18·57-s + 3·59-s + 3·61-s − 6·67-s − 9·69-s + 9·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.68·5-s + 1.13·7-s + 1.80·11-s − 4.64·15-s − 1.37·19-s + 1.96·21-s − 0.625·23-s + 12/5·25-s − 1.92·27-s + 0.557·29-s + 2.15·31-s + 3.13·33-s − 3.04·35-s + 2.95·37-s + 1.40·41-s + 2.28·43-s − 0.875·47-s + 6/7·49-s − 1.23·53-s − 4.85·55-s − 2.38·57-s + 0.390·59-s + 0.384·61-s − 0.733·67-s − 1.08·69-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.477028841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.477028841\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 17 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 17 T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_j_ar |
| 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 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} - 5 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_d_af |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 135 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bq_aff |
| 13 | $A_4\times C_2$ | \( 1 + 18 T^{2} - 17 T^{3} + 18 p T^{4} + p^{3} T^{6} \) | 3.13.a_s_ar |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 12 T^{2} + 15 T^{3} + 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_m_p |
| 23 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 81 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.d_bh_dd |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 175 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_dd_agt |
| 31 | $A_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 763 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.am_fc_abdj |
| 37 | $A_4\times C_2$ | \( 1 - 18 T + 210 T^{2} - 1503 T^{3} + 210 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.as_ic_acfv |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 657 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aj_et_azh |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 201 T^{2} - 1399 T^{3} + 201 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ap_ht_acbv |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 6 T^{2} - 405 T^{3} + 6 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_g_app |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 631 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_eh_yh |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 171 T^{2} - 355 T^{3} + 171 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_gp_anr |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 403 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_cr_apn |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} - 565 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_y_avt |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 165 T^{2} - 955 T^{3} + 165 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_gj_abkt |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 23 p T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_iu_acmp |
| 79 | $A_4\times C_2$ | \( 1 + 6 T + 213 T^{2} + 956 T^{3} + 213 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_if_bku |
| 83 | $A_4\times C_2$ | \( 1 - 21 T + 168 T^{2} - 929 T^{3} + 168 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.av_gm_abjt |
| 89 | $A_4\times C_2$ | \( 1 - 18 T + 282 T^{2} - 2573 T^{3} + 282 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.as_kw_aduz |
| 97 | $A_4\times C_2$ | \( 1 + 9 T + 126 T^{2} + 1709 T^{3} + 126 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.j_ew_cnt |
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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
−8.223643372291364073236295807563, −7.81537664639841148355023087319, −7.56826342795700084504149190679, −7.40733769442378881782269628460, −7.35883860346309626778924941310, −6.60367725861163862771312751360, −6.47118558549801376379306928655, −6.10737174437908947971314498185, −6.10593748296852025768368226020, −5.76048311415198377964984642790, −5.08635483207157963322011599289, −4.72294047751615718051807026408, −4.56275392108585391434902381977, −4.18832022841204745352538963845, −4.13688395343795041819908969218, −3.99558442700873346399679240889, −3.51375624921067927237851848481, −3.20879679292269372006579947405, −3.10478991633239804757630107476, −2.41182119028116093188579519810, −2.23514938821122200010316856570, −2.21064990214678319353775122227, −1.27634970687635317308762488667, −0.801578779066155387839762109186, −0.55848886095697809192972880237,
0.55848886095697809192972880237, 0.801578779066155387839762109186, 1.27634970687635317308762488667, 2.21064990214678319353775122227, 2.23514938821122200010316856570, 2.41182119028116093188579519810, 3.10478991633239804757630107476, 3.20879679292269372006579947405, 3.51375624921067927237851848481, 3.99558442700873346399679240889, 4.13688395343795041819908969218, 4.18832022841204745352538963845, 4.56275392108585391434902381977, 4.72294047751615718051807026408, 5.08635483207157963322011599289, 5.76048311415198377964984642790, 6.10593748296852025768368226020, 6.10737174437908947971314498185, 6.47118558549801376379306928655, 6.60367725861163862771312751360, 7.35883860346309626778924941310, 7.40733769442378881782269628460, 7.56826342795700084504149190679, 7.81537664639841148355023087319, 8.223643372291364073236295807563
