| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 9·6-s + 2·7-s − 10·8-s + 6·9-s − 4·11-s + 18·12-s − 6·14-s + 15·16-s + 4·17-s − 18·18-s − 6·19-s + 6·21-s + 12·22-s − 4·23-s − 30·24-s + 10·27-s + 12·28-s − 12·31-s − 21·32-s − 12·33-s − 12·34-s + 36·36-s − 22·37-s + 18·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 3.67·6-s + 0.755·7-s − 3.53·8-s + 2·9-s − 1.20·11-s + 5.19·12-s − 1.60·14-s + 15/4·16-s + 0.970·17-s − 4.24·18-s − 1.37·19-s + 1.30·21-s + 2.55·22-s − 0.834·23-s − 6.12·24-s + 1.92·27-s + 2.26·28-s − 2.15·31-s − 3.71·32-s − 2.08·33-s − 2.05·34-s + 6·36-s − 3.61·37-s + 2.91·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_r_ay |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 68 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bd_cq |
| 13 | $S_4\times C_2$ | \( 1 + 23 T^{2} + 16 T^{3} + 23 p T^{4} + p^{3} T^{6} \) | 3.13.a_x_q |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 128 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_bp_ey |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 152 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_cb_fw |
| 29 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 20 T^{3} + 39 p T^{4} + p^{3} T^{6} \) | 3.29.a_bn_u |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 424 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.m_cz_qi |
| 37 | $D_{6}$ | \( 1 + 22 T + 7 p T^{2} + 52 p T^{3} + 7 p^{2} T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.w_jz_cwa |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 784 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.q_ez_bee |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 324 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ae_bh_amm |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 191 T^{2} - 1568 T^{3} + 191 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aq_hj_acii |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.59.g_hh_bbo |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{3} \) | 3.61.be_sp_gxg |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 185 T^{2} - 912 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ai_hd_abjc |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 233 T^{2} + 1412 T^{3} + 233 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_iz_cci |
| 73 | $S_4\times C_2$ | \( 1 + 107 T^{2} + 416 T^{3} + 107 p T^{4} + p^{3} T^{6} \) | 3.73.a_ed_qa |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 377 T^{2} + 3972 T^{3} + 377 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ba_on_fwu |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) | 3.83.abk_baf_alki |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 279 T^{2} + 1740 T^{3} + 279 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.k_kt_coy |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 99 T^{2} - 496 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.c_dv_atc |
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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
−7.55640668074182451353814169238, −7.38388097872615442423265770199, −7.29422850500036366674610340470, −7.05680076196159295219419750714, −6.51347055223652645225589755235, −6.49632430201118236769457959819, −6.46488514738700540403001562338, −5.77542163139848399357967420676, −5.59850032332676776150721460549, −5.46556876375901529887840061733, −4.97536214570501854888523998144, −4.86999534925183579536097431760, −4.69277105462349452303577905662, −4.04981340752374364979818094154, −3.87205191242702474019940522474, −3.56428419981056456803942095403, −3.28159090182549919421163036047, −3.20999757242948771572713092983, −2.84366829292612123083832402344, −2.29562795184485398681094632257, −2.26246717440897507064204896666, −1.97765817277161069886914678489, −1.54894020260476096756564921400, −1.41840667205584049754958357349, −1.31313452405800214870445621919, 0, 0, 0,
1.31313452405800214870445621919, 1.41840667205584049754958357349, 1.54894020260476096756564921400, 1.97765817277161069886914678489, 2.26246717440897507064204896666, 2.29562795184485398681094632257, 2.84366829292612123083832402344, 3.20999757242948771572713092983, 3.28159090182549919421163036047, 3.56428419981056456803942095403, 3.87205191242702474019940522474, 4.04981340752374364979818094154, 4.69277105462349452303577905662, 4.86999534925183579536097431760, 4.97536214570501854888523998144, 5.46556876375901529887840061733, 5.59850032332676776150721460549, 5.77542163139848399357967420676, 6.46488514738700540403001562338, 6.49632430201118236769457959819, 6.51347055223652645225589755235, 7.05680076196159295219419750714, 7.29422850500036366674610340470, 7.38388097872615442423265770199, 7.55640668074182451353814169238
