| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 3·5-s + 9·6-s − 6·7-s + 10·8-s + 6·9-s + 9·10-s − 12·11-s + 18·12-s − 18·14-s + 9·15-s + 15·16-s + 18·18-s + 3·19-s + 18·20-s − 18·21-s − 36·22-s − 6·23-s + 30·24-s + 6·25-s + 10·27-s − 36·28-s − 6·29-s + 27·30-s − 18·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s − 2.26·7-s + 3.53·8-s + 2·9-s + 2.84·10-s − 3.61·11-s + 5.19·12-s − 4.81·14-s + 2.32·15-s + 15/4·16-s + 4.24·18-s + 0.688·19-s + 4.02·20-s − 3.92·21-s − 7.67·22-s − 1.25·23-s + 6.12·24-s + 6/5·25-s + 1.92·27-s − 6.80·28-s − 1.11·29-s + 4.92·30-s − 3.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 17 | | \( 1 \) | |
| good | 7 | $A_4\times C_2$ | \( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.g_be_dh |
| 11 | $A_4\times C_2$ | \( 1 + 12 T + 78 T^{2} + 315 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.m_da_md |
| 13 | $A_4\times C_2$ | \( 1 + 18 T^{2} + 37 T^{3} + 18 p T^{4} + p^{3} T^{6} \) | 3.13.a_s_bl |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 131 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_m_afb |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_ci_ir |
| 29 | $A_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 324 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_dj_mm |
| 31 | $A_4\times C_2$ | \( 1 + 18 T + 189 T^{2} + 1252 T^{3} + 189 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.s_hh_bwe |
| 37 | $A_4\times C_2$ | \( 1 + 9 T + 126 T^{2} + 649 T^{3} + 126 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.j_ew_yz |
| 41 | $A_4\times C_2$ | \( 1 + 15 T + 186 T^{2} + 1287 T^{3} + 186 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.p_he_bxn |
| 43 | $A_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 508 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.g_ez_to |
| 47 | $A_4\times C_2$ | \( 1 + 12 T + 150 T^{2} + 1125 T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_fu_brh |
| 53 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_fu_yp |
| 59 | $A_4\times C_2$ | \( 1 + 114 T^{2} + 171 T^{3} + 114 p T^{4} + p^{3} T^{6} \) | 3.59.a_ek_gp |
| 61 | $A_4\times C_2$ | \( 1 + 18 T + 243 T^{2} + 2188 T^{3} + 243 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.s_jj_dge |
| 67 | $A_4\times C_2$ | \( 1 + 30 T + 465 T^{2} + 4588 T^{3} + 465 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.be_rx_gum |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 33 T^{2} - 36 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.g_bh_abk |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 123 T^{2} - 884 T^{3} + 123 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_et_abia |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 213 T^{2} - 812 T^{3} + 213 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ag_if_abfg |
| 83 | $A_4\times C_2$ | \( 1 - 24 T + 393 T^{2} - 4176 T^{3} + 393 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ay_pd_ageq |
| 89 | $A_4\times C_2$ | \( 1 - 6 T - 12 T^{2} + 1359 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_am_cah |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 195 T^{2} - 740 T^{3} + 195 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ag_hn_abcm |
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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
−7.25240154062861739128609635167, −6.76066488858824997617386863015, −6.73035732401241461956626364467, −6.51617689846837552082626870836, −6.09867695038230285139748238221, −6.00971124922082683876903359119, −5.88124580487958956977993918886, −5.50700955846678294761091267585, −5.23170418220714589345411057935, −5.13293599522968322518527895896, −4.87234883242456991378946785112, −4.71306168099882662855335340384, −4.58471242012676235308681703575, −3.72733969620528955625706042943, −3.64465619619727052625494431202, −3.58428389503713129395055574156, −3.17587440999725200873130176752, −3.16702207742172259158641565232, −3.14329741361790742666346307189, −2.49271134149060622854396381951, −2.44011283825417396580441599937, −2.21175102197035264728145642275, −1.75086007167843412180059835427, −1.59015653753308916951349240472, −1.54898806303750532261549560074, 0, 0, 0,
1.54898806303750532261549560074, 1.59015653753308916951349240472, 1.75086007167843412180059835427, 2.21175102197035264728145642275, 2.44011283825417396580441599937, 2.49271134149060622854396381951, 3.14329741361790742666346307189, 3.16702207742172259158641565232, 3.17587440999725200873130176752, 3.58428389503713129395055574156, 3.64465619619727052625494431202, 3.72733969620528955625706042943, 4.58471242012676235308681703575, 4.71306168099882662855335340384, 4.87234883242456991378946785112, 5.13293599522968322518527895896, 5.23170418220714589345411057935, 5.50700955846678294761091267585, 5.88124580487958956977993918886, 6.00971124922082683876903359119, 6.09867695038230285139748238221, 6.51617689846837552082626870836, 6.73035732401241461956626364467, 6.76066488858824997617386863015, 7.25240154062861739128609635167
