| L(s) = 1 | + 3·2-s + 6·4-s + 6·5-s + 6·7-s + 10·8-s − 6·9-s + 18·10-s + 6·11-s − 6·13-s + 18·14-s + 15·16-s + 3·17-s − 18·18-s + 36·20-s + 18·22-s + 9·25-s − 18·26-s − 27-s + 36·28-s + 12·29-s − 6·31-s + 21·32-s + 9·34-s + 36·35-s − 36·36-s − 6·37-s + 60·40-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 2.68·5-s + 2.26·7-s + 3.53·8-s − 2·9-s + 5.69·10-s + 1.80·11-s − 1.66·13-s + 4.81·14-s + 15/4·16-s + 0.727·17-s − 4.24·18-s + 8.04·20-s + 3.83·22-s + 9/5·25-s − 3.53·26-s − 0.192·27-s + 6.80·28-s + 2.22·29-s − 1.07·31-s + 3.71·32-s + 1.54·34-s + 6.08·35-s − 6·36-s − 0.986·37-s + 9.48·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.80063010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.80063010\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) | 3.3.a_g_b |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.5.ag_bb_acq |
| 7 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 60 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ag_v_aci |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 113 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bk_aej |
| 13 | $A_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 148 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bn_fs |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 9 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_g_j |
| 23 | $A_4\times C_2$ | \( 1 + 57 T^{2} - 8 T^{3} + 57 p T^{4} + p^{3} T^{6} \) | 3.23.a_cf_ai |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 99 T^{2} - 544 T^{3} + 99 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_dv_auy |
| 31 | $A_4\times C_2$ | \( 1 + 6 T + 69 T^{2} + 380 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.g_cr_oq |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 308 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_dj_lw |
| 41 | $A_4\times C_2$ | \( 1 + 120 T^{2} - T^{3} + 120 p T^{4} + p^{3} T^{6} \) | 3.41.a_eq_ab |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 757 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_fo_abdd |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 69 T^{2} + 268 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_cr_ki |
| 53 | $A_4\times C_2$ | \( 1 + 75 T^{2} + 136 T^{3} + 75 p T^{4} + p^{3} T^{6} \) | 3.53.a_cx_fg |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 711 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.g_gm_bbj |
| 61 | $A_4\times C_2$ | \( 1 - 18 T + 279 T^{2} - 2348 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.as_kt_admi |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 282 T^{2} + 2493 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.s_kw_drx |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 225 T^{2} + 1552 T^{3} + 225 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_ir_chs |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 1695 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_ga_acnf |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 333 T^{2} + 2980 T^{3} + 333 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.s_mv_ekq |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ag_io_abkj |
| 89 | $A_4\times C_2$ | \( 1 - 9 T + 150 T^{2} - 621 T^{3} + 150 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aj_fu_axx |
| 97 | $A_4\times C_2$ | \( 1 + 288 T^{2} - T^{3} + 288 p T^{4} + p^{3} T^{6} \) | 3.97.a_lc_ab |
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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
−9.384190651795457235607134062847, −8.896210656722049157898024401297, −8.816199520326180757740212528776, −8.473229390319631569558225690595, −7.75088044867944126053004607875, −7.74027486224275091101613003797, −7.68970678618846764847856019015, −6.89188638082613117449474092731, −6.55317285559256045478504321872, −6.54724719260908931547577882500, −5.96037711032682138094873566860, −5.82508576535378082352992040760, −5.64050138065177359492559229806, −5.15227116764497777323788435945, −5.14834280382561927379914243387, −4.87453971753076875163518206284, −4.39646983391725764260083786340, −3.97749478040199498885872469865, −3.66345535925554804191450247479, −2.98402601872513900500463677113, −2.55918184895952397256267162215, −2.49807224975882712898764232513, −1.95072009451188635837293216049, −1.51728678559143877194733992712, −1.39036145962278626257464958440,
1.39036145962278626257464958440, 1.51728678559143877194733992712, 1.95072009451188635837293216049, 2.49807224975882712898764232513, 2.55918184895952397256267162215, 2.98402601872513900500463677113, 3.66345535925554804191450247479, 3.97749478040199498885872469865, 4.39646983391725764260083786340, 4.87453971753076875163518206284, 5.14834280382561927379914243387, 5.15227116764497777323788435945, 5.64050138065177359492559229806, 5.82508576535378082352992040760, 5.96037711032682138094873566860, 6.54724719260908931547577882500, 6.55317285559256045478504321872, 6.89188638082613117449474092731, 7.68970678618846764847856019015, 7.74027486224275091101613003797, 7.75088044867944126053004607875, 8.473229390319631569558225690595, 8.816199520326180757740212528776, 8.896210656722049157898024401297, 9.384190651795457235607134062847
