| L(s) = 1 | + 3·3-s + 3·5-s + 6·9-s + 6·11-s + 9·15-s + 2·17-s + 12·19-s − 2·23-s + 6·25-s + 10·27-s + 4·29-s − 3·31-s + 18·33-s − 6·37-s + 12·41-s + 6·43-s + 18·45-s + 4·47-s − 5·49-s + 6·51-s + 2·53-s + 18·55-s + 36·57-s + 14·59-s + 2·61-s − 2·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 2·9-s + 1.80·11-s + 2.32·15-s + 0.485·17-s + 2.75·19-s − 0.417·23-s + 6/5·25-s + 1.92·27-s + 0.742·29-s − 0.538·31-s + 3.13·33-s − 0.986·37-s + 1.87·41-s + 0.914·43-s + 2.68·45-s + 0.583·47-s − 5/7·49-s + 0.840·51-s + 0.274·53-s + 2.42·55-s + 4.76·57-s + 1.82·59-s + 0.256·61-s − 0.244·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.01644150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.01644150\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 18 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.7.a_f_s |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.11.ag_bt_afk |
| 13 | $S_4\times C_2$ | \( 1 + 23 T^{2} - 18 T^{3} + 23 p T^{4} + p^{3} T^{6} \) | 3.13.a_x_as |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 11 T^{2} - 24 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_l_ay |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.19.am_eb_aua |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 29 T^{2} + 48 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_bd_bw |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 286 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ae_bt_ala |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 402 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_ed_pm |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.41.am_gp_aboi |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.43.ag_fl_aue |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 244 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ae_cj_ajk |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 75 T^{2} - 176 T^{3} + 75 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ac_cx_agu |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 195 T^{2} - 1418 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ao_hn_acco |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 268 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ac_t_aki |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + T^{2} - 554 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.c_b_avi |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 63 T^{2} - 134 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_cl_afe |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 1082 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_dj_abpq |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 117 T^{2} - 20 T^{3} + 117 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_en_au |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 241 T^{2} + 1576 T^{3} + 241 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.k_jh_ciq |
| 89 | $S_4\times C_2$ | \( 1 - 34 T + 585 T^{2} - 6538 T^{3} + 585 p T^{4} - 34 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abi_wn_ajrm |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 147 T^{2} - 1264 T^{3} + 147 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ai_fr_abwq |
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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
−8.351969417643396947550714089297, −7.72174215922449424876916327041, −7.66257684755500477486266978727, −7.62796533625559279231104580364, −7.15212084999887000889294162845, −6.94823420324601439322267913734, −6.71844871281913631494446307180, −6.21355587853058057737994098734, −6.15574709724955192840857649739, −5.86271365963368734476574782844, −5.36118068344479225126909057859, −5.22441304327378509086690708100, −4.99371141178401521139424909365, −4.39948425248671197545840212294, −4.27377455497184755447851439605, −3.83223300507774775707693070471, −3.47417553242543839160491035304, −3.29622285562195212130302604659, −3.17901469621850768464261458095, −2.39366358737741486426872338392, −2.29918589076365180279408763402, −2.14113397848903863221795863373, −1.27229412037878514002895456979, −1.16246646761617341493026923609, −1.00902756371024399298149263107,
1.00902756371024399298149263107, 1.16246646761617341493026923609, 1.27229412037878514002895456979, 2.14113397848903863221795863373, 2.29918589076365180279408763402, 2.39366358737741486426872338392, 3.17901469621850768464261458095, 3.29622285562195212130302604659, 3.47417553242543839160491035304, 3.83223300507774775707693070471, 4.27377455497184755447851439605, 4.39948425248671197545840212294, 4.99371141178401521139424909365, 5.22441304327378509086690708100, 5.36118068344479225126909057859, 5.86271365963368734476574782844, 6.15574709724955192840857649739, 6.21355587853058057737994098734, 6.71844871281913631494446307180, 6.94823420324601439322267913734, 7.15212084999887000889294162845, 7.62796533625559279231104580364, 7.66257684755500477486266978727, 7.72174215922449424876916327041, 8.351969417643396947550714089297
