| L(s) = 1 | + 3-s + 5-s + 9·7-s − 3·9-s + 7·11-s + 4·13-s + 15-s + 19-s + 9·21-s − 23-s + 25-s + 27-s + 2·31-s + 7·33-s + 9·35-s + 2·37-s + 4·39-s − 13·41-s − 6·43-s − 3·45-s − 7·47-s + 33·49-s − 12·53-s + 7·55-s + 57-s + 23·59-s + 7·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 3.40·7-s − 9-s + 2.11·11-s + 1.10·13-s + 0.258·15-s + 0.229·19-s + 1.96·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.359·31-s + 1.21·33-s + 1.52·35-s + 0.328·37-s + 0.640·39-s − 2.03·41-s − 0.914·43-s − 0.447·45-s − 1.02·47-s + 33/7·49-s − 1.64·53-s + 0.943·55-s + 0.132·57-s + 2.99·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.21173006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.21173006\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 29 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 4 T^{2} - 8 T^{3} + 4 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_e_ai |
| 5 | $S_4\times C_2$ | \( 1 - T + 18 T^{3} - p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ab_a_s |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) | 3.7.aj_bw_afx |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 2 p T^{2} - 48 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ah_w_abw |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 38 T^{2} - 96 T^{3} + 38 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_bm_ads |
| 17 | $S_4\times C_2$ | \( 1 + 3 T^{2} + 101 T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.17.a_d_dx |
| 19 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 6 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ab_bm_ag |
| 23 | $S_4\times C_2$ | \( 1 + T + 19 T^{2} - 102 T^{3} + 19 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.23.b_t_ady |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 89 T^{2} - 117 T^{3} + 89 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ac_dl_aen |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 22 T^{2} - 206 T^{3} + 22 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ac_w_ahy |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 122 T^{2} + 812 T^{3} + 122 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.n_es_bfg |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 125 T^{2} + 484 T^{3} + 125 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.g_ev_sq |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 96 T^{2} + 711 T^{3} + 96 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.h_ds_bbj |
| 53 | $S_4\times C_2$ | \( 1 + 12 T + 35 T^{2} - 200 T^{3} + 35 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.m_bj_ahs |
| 59 | $S_4\times C_2$ | \( 1 - 23 T + 314 T^{2} - 2776 T^{3} + 314 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ax_mc_aecu |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 420 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ah_cq_aqe |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 137 T^{2} - 624 T^{3} + 137 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ai_fh_aya |
| 71 | $S_4\times C_2$ | \( 1 - 3 T + 113 T^{2} + 70 T^{3} + 113 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ad_ej_cs |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 333 T^{2} - 2986 T^{3} + 333 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.at_mv_aekw |
| 79 | $S_4\times C_2$ | \( 1 + 24 T + 383 T^{2} + 3989 T^{3} + 383 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.y_ot_fxl |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 220 T^{2} - 1344 T^{3} + 220 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ai_im_abzs |
| 89 | $S_4\times C_2$ | \( 1 - 5 T + 254 T^{2} - 889 T^{3} + 254 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.af_ju_abif |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 173 T^{2} + 2181 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.k_gr_dfx |
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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
−7.13507450754650667466993360507, −6.60099591535714399907404969671, −6.60089236254815274590840060100, −6.49770001012858527460580583436, −6.11133377719599731030296960084, −5.93245997107224591928691144497, −5.51692370324248926677894789687, −5.29760355791334823006449138599, −4.94738770456320264223859463625, −4.93394084211376152125547627014, −4.82431703067006726356439382471, −4.50455617213950028651535660842, −3.99131324539128784241882918837, −3.84359328661838417048869795001, −3.61379603257345438936620822168, −3.48354430996843036354577971126, −3.03035441732665710157572130610, −2.63922692443319580596555029765, −2.39565163015770440055224080599, −1.96064287633079717399283447786, −1.67635575432114190024399603978, −1.58704772459137421131347112340, −1.38729236145873465787238479983, −0.843736697529928884895368727519, −0.61924253530480637502028368099,
0.61924253530480637502028368099, 0.843736697529928884895368727519, 1.38729236145873465787238479983, 1.58704772459137421131347112340, 1.67635575432114190024399603978, 1.96064287633079717399283447786, 2.39565163015770440055224080599, 2.63922692443319580596555029765, 3.03035441732665710157572130610, 3.48354430996843036354577971126, 3.61379603257345438936620822168, 3.84359328661838417048869795001, 3.99131324539128784241882918837, 4.50455617213950028651535660842, 4.82431703067006726356439382471, 4.93394084211376152125547627014, 4.94738770456320264223859463625, 5.29760355791334823006449138599, 5.51692370324248926677894789687, 5.93245997107224591928691144497, 6.11133377719599731030296960084, 6.49770001012858527460580583436, 6.60089236254815274590840060100, 6.60099591535714399907404969671, 7.13507450754650667466993360507
