| L(s) = 1 | − 3·2-s + 3·4-s + 3·5-s + 3·7-s − 9·10-s + 9·11-s − 3·13-s − 9·14-s − 3·16-s + 3·19-s + 9·20-s − 27·22-s − 6·23-s + 3·25-s + 9·26-s − 9·27-s + 9·28-s + 3·29-s + 9·31-s + 6·32-s + 9·35-s + 6·37-s − 9·38-s − 3·41-s − 6·43-s + 27·44-s + 18·46-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.34·5-s + 1.13·7-s − 2.84·10-s + 2.71·11-s − 0.832·13-s − 2.40·14-s − 3/4·16-s + 0.688·19-s + 2.01·20-s − 5.75·22-s − 1.25·23-s + 3/5·25-s + 1.76·26-s − 1.73·27-s + 1.70·28-s + 0.557·29-s + 1.61·31-s + 1.06·32-s + 1.52·35-s + 0.986·37-s − 1.45·38-s − 0.468·41-s − 0.914·43-s + 4.07·44-s + 2.65·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171588264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171588264\) |
| \(L(\frac{3}{2})\) |
|
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 | 17 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.2.d_g_j |
| 3 | $C_6$ | \( 1 + p^{2} T^{3} + p^{3} T^{6} \) | 3.3.a_a_j |
| 5 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 11 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_g_al |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 23 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_m_ax |
| 11 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 215 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.aj_cf_aih |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 21 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_v_v |
| 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 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 171 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_dj_agp |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_dp_aul |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 427 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dy_aql |
| 41 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} - 243 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.d_av_ajj |
| 43 | $A_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 220 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.g_cf_im |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 333 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_cr_mv |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 102 T^{2} + 433 T^{3} + 102 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_dy_qr |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 844 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.g_eb_bgm |
| 61 | $A_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 639 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.aj_ez_ayp |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.67.g_if_bfg |
| 71 | $A_4\times C_2$ | \( 1 + 102 T^{2} - 323 T^{3} + 102 p T^{4} + p^{3} T^{6} \) | 3.71.a_dy_aml |
| 73 | $A_4\times C_2$ | \( 1 - 30 T + 516 T^{2} - 5351 T^{3} + 516 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.abe_tw_ahxv |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 1345 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ag_dy_abzt |
| 83 | $A_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 1584 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.am_fl_aciy |
| 89 | $A_4\times C_2$ | \( 1 - 9 T + 165 T^{2} - 1313 T^{3} + 165 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aj_gj_abyn |
| 97 | $A_4\times C_2$ | \( 1 + 24 T + 447 T^{2} + 4952 T^{3} + 447 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.y_rf_him |
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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
−7.43930475134759987884130664382, −7.08628107958793826881833371661, −6.56666522143584353544040310031, −6.54200533651531659702486277030, −6.32554946882724584486169838692, −6.30941713946887343733207031528, −5.87266685569789245541663872319, −5.68105302708791464964135558089, −5.15102837050990086296186856858, −5.00643414120219671051475051455, −4.75001716014625243146639448145, −4.70635490382963145317872188231, −4.23512042685964750580016123309, −3.76802580393743982739996057098, −3.68879888132707942192160407406, −3.61227568702175487971356478679, −2.92076844782660563619516182536, −2.60754881558921047674079870060, −2.20502773656028934119631988610, −2.00984422493198948012556555798, −1.72201102724492911750360482829, −1.29928004614761911498607768681, −1.25710549849800145540079673168, −0.800028448995094969343354003746, −0.30639654477353588480080287840,
0.30639654477353588480080287840, 0.800028448995094969343354003746, 1.25710549849800145540079673168, 1.29928004614761911498607768681, 1.72201102724492911750360482829, 2.00984422493198948012556555798, 2.20502773656028934119631988610, 2.60754881558921047674079870060, 2.92076844782660563619516182536, 3.61227568702175487971356478679, 3.68879888132707942192160407406, 3.76802580393743982739996057098, 4.23512042685964750580016123309, 4.70635490382963145317872188231, 4.75001716014625243146639448145, 5.00643414120219671051475051455, 5.15102837050990086296186856858, 5.68105302708791464964135558089, 5.87266685569789245541663872319, 6.30941713946887343733207031528, 6.32554946882724584486169838692, 6.54200533651531659702486277030, 6.56666522143584353544040310031, 7.08628107958793826881833371661, 7.43930475134759987884130664382
