| L(s) = 1 | − 2·5-s − 5·7-s + 3·11-s + 9·13-s + 17-s + 3·19-s − 9·23-s + 3·25-s − 10·29-s − 6·31-s + 10·35-s − 3·37-s − 2·41-s + 12·43-s − 10·47-s + 12·49-s − 19·53-s − 6·55-s + 10·59-s + 14·61-s − 18·65-s + 14·67-s + 16·71-s + 9·73-s − 15·77-s + 2·79-s − 13·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.88·7-s + 0.904·11-s + 2.49·13-s + 0.242·17-s + 0.688·19-s − 1.87·23-s + 3/5·25-s − 1.85·29-s − 1.07·31-s + 1.69·35-s − 0.493·37-s − 0.312·41-s + 1.82·43-s − 1.45·47-s + 12/7·49-s − 2.60·53-s − 0.809·55-s + 1.30·59-s + 1.79·61-s − 2.23·65-s + 1.71·67-s + 1.89·71-s + 1.05·73-s − 1.70·77-s + 0.225·79-s − 1.42·83-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
| \(L(1)\) |
\(\approx\) |
\(1.403491588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403491588\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + 2 T + T^{2} - 12 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_b_am |
| 7 | $S_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 26 T^{3} + 13 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.f_n_ba |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 29 T^{2} - 58 T^{3} + 29 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ad_bd_acg |
| 13 | $S_4\times C_2$ | \( 1 - 9 T + 59 T^{2} - 238 T^{3} + 59 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.aj_ch_aje |
| 17 | $S_4\times C_2$ | \( 1 - T + 11 T^{2} - 80 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ab_l_adc |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 53 T^{2} - 106 T^{3} + 53 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_cb_aec |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 53 T^{2} + 268 T^{3} + 53 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.j_cb_ki |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 105 T^{2} + 588 T^{3} + 105 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.k_eb_wq |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.g_cz_lw |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} - 132 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_h_afc |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.43.am_gv_abqe |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 109 T^{2} + 588 T^{3} + 109 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.k_ef_wq |
| 53 | $S_4\times C_2$ | \( 1 + 19 T + 219 T^{2} + 1722 T^{3} + 219 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.t_il_cog |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 131 T^{2} - 1052 T^{3} + 131 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ak_fb_abom |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 131 T^{2} - 1044 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_fb_aboe |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 197 T^{2} - 1692 T^{3} + 197 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ao_hp_acnc |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 229 T^{2} - 2016 T^{3} + 229 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_iv_aczo |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 183 T^{2} - 1098 T^{3} + 183 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_hb_abqg |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 141 T^{2} - 620 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_fl_axw |
| 83 | $S_4\times C_2$ | \( 1 + 13 T + 245 T^{2} + 1814 T^{3} + 245 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.n_jl_cru |
| 89 | $S_4\times C_2$ | \( 1 - 25 T + 435 T^{2} - 4752 T^{3} + 435 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.az_qt_ahau |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 391 T^{2} - 4036 T^{3} + 391 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.aw_pb_afzg |
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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
−7.981174090863176581477406956312, −7.45299923613510577383485455653, −7.36859119271711698930199125486, −7.10511824545202553525385257276, −6.77604814335516858221182206136, −6.44596751160988432953248423031, −6.34940988154778453809160058227, −6.10625742110567292714159780552, −5.84523666257481041073025258636, −5.63661165364768095607951056588, −5.35155665707522967373744547951, −4.85944080983411173745651860957, −4.66429902530916831336868071483, −4.01109223502706989167886184496, −3.98326032915654709360328656025, −3.67090592935885855528197104280, −3.40038422195508452218928012879, −3.36127522853929821606852034702, −3.28991443615811498682303877789, −2.35684253513855613635700137385, −1.99908483700380751262289046642, −1.94212126782564099156970287715, −1.13509590443831953241932997957, −0.862161620629818326766251517482, −0.31650061998869860534323432221,
0.31650061998869860534323432221, 0.862161620629818326766251517482, 1.13509590443831953241932997957, 1.94212126782564099156970287715, 1.99908483700380751262289046642, 2.35684253513855613635700137385, 3.28991443615811498682303877789, 3.36127522853929821606852034702, 3.40038422195508452218928012879, 3.67090592935885855528197104280, 3.98326032915654709360328656025, 4.01109223502706989167886184496, 4.66429902530916831336868071483, 4.85944080983411173745651860957, 5.35155665707522967373744547951, 5.63661165364768095607951056588, 5.84523666257481041073025258636, 6.10625742110567292714159780552, 6.34940988154778453809160058227, 6.44596751160988432953248423031, 6.77604814335516858221182206136, 7.10511824545202553525385257276, 7.36859119271711698930199125486, 7.45299923613510577383485455653, 7.981174090863176581477406956312
