| L(s) = 1 | + 3·5-s − 6·11-s − 9·17-s + 15·19-s + 6·23-s + 6·25-s − 3·31-s + 9·37-s − 3·41-s + 3·43-s − 6·47-s − 9·49-s + 21·53-s − 18·55-s + 9·59-s − 12·61-s + 6·67-s − 9·71-s − 3·73-s + 12·79-s + 21·83-s − 27·85-s + 6·89-s + 45·95-s + 18·97-s + 9·101-s + 6·107-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.80·11-s − 2.18·17-s + 3.44·19-s + 1.25·23-s + 6/5·25-s − 0.538·31-s + 1.47·37-s − 0.468·41-s + 0.457·43-s − 0.875·47-s − 9/7·49-s + 2.88·53-s − 2.42·55-s + 1.17·59-s − 1.53·61-s + 0.733·67-s − 1.06·71-s − 0.351·73-s + 1.35·79-s + 2.30·83-s − 2.92·85-s + 0.635·89-s + 4.61·95-s + 1.82·97-s + 0.895·101-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.857122115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.857122115\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 12 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.7.a_j_am |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 104 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bh_ea |
| 13 | $S_4\times C_2$ | \( 1 + 27 T^{2} - 12 T^{3} + 27 p T^{4} + p^{3} T^{6} \) | 3.13.a_bb_am |
| 17 | $S_4\times C_2$ | \( 1 + 9 T + 66 T^{2} + 299 T^{3} + 66 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.j_co_ln |
| 19 | $S_4\times C_2$ | \( 1 - 15 T + 120 T^{2} - 623 T^{3} + 120 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ap_eq_axz |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 280 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_cf_aku |
| 29 | $S_4\times C_2$ | \( 1 + 63 T^{2} + 44 T^{3} + 63 p T^{4} + p^{3} T^{6} \) | 3.29.a_cl_bs |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 114 T^{2} - 595 T^{3} + 114 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.aj_ek_awx |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 279 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.d_be_kt |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 151 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ad_a_fv |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 129 T^{2} + 540 T^{3} + 129 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_ez_uu |
| 53 | $S_4\times C_2$ | \( 1 - 21 T + 240 T^{2} - 1969 T^{3} + 240 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.av_jg_acxt |
| 59 | $S_4\times C_2$ | \( 1 - 9 T + 180 T^{2} - 1033 T^{3} + 180 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.aj_gy_abnt |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 207 T^{2} + 1416 T^{3} + 207 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_hz_ccm |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 448 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_ez_arg |
| 71 | $S_4\times C_2$ | \( 1 + 9 T + 48 T^{2} + 381 T^{3} + 48 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.j_bw_or |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 48 T^{2} + 1131 T^{3} + 48 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.d_bw_brn |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 153 T^{2} - 884 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_fx_abia |
| 83 | $S_4\times C_2$ | \( 1 - 21 T + 372 T^{2} - 3687 T^{3} + 372 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.av_oi_aflv |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 147 T^{2} - 264 T^{3} + 147 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_fr_ake |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 351 T^{2} - 3436 T^{3} + 351 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.as_nn_afce |
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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
−7.26484677623278657581229124571, −6.88881665134887644919679099628, −6.76410557625785273767209767913, −6.62486567994022916777338357478, −6.04899751438973978736827984741, −5.92744843546304665725006024603, −5.87897905894575108056997151763, −5.32397148269162671482457755682, −5.23283565954227082652497866450, −5.22366864669067617490863438283, −4.72376988570513983343946201232, −4.66387532670382535954291032301, −4.44288951416009400071472282190, −3.91401471280609171991949629270, −3.43779065386458505822476125646, −3.41496801494461273186538162028, −2.93109198346281683927474444814, −2.84566998098669620242719392953, −2.61527627227805696611540185601, −2.05693079361970857190069189241, −1.91888402797117088422001016759, −1.81411434120040261740162351480, −0.991935267736165126764160471228, −0.72103317585338146105835372238, −0.56820269907648783376313704662,
0.56820269907648783376313704662, 0.72103317585338146105835372238, 0.991935267736165126764160471228, 1.81411434120040261740162351480, 1.91888402797117088422001016759, 2.05693079361970857190069189241, 2.61527627227805696611540185601, 2.84566998098669620242719392953, 2.93109198346281683927474444814, 3.41496801494461273186538162028, 3.43779065386458505822476125646, 3.91401471280609171991949629270, 4.44288951416009400071472282190, 4.66387532670382535954291032301, 4.72376988570513983343946201232, 5.22366864669067617490863438283, 5.23283565954227082652497866450, 5.32397148269162671482457755682, 5.87897905894575108056997151763, 5.92744843546304665725006024603, 6.04899751438973978736827984741, 6.62486567994022916777338357478, 6.76410557625785273767209767913, 6.88881665134887644919679099628, 7.26484677623278657581229124571
