| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s − 3·5-s + 9·6-s + 3·7-s − 10·8-s + 3·9-s + 9·10-s − 18·12-s + 3·13-s − 9·14-s + 9·15-s + 15·16-s − 3·17-s − 9·18-s − 18·20-s − 9·21-s − 15·23-s + 30·24-s + 6·25-s − 9·26-s − 27-s + 18·28-s − 27·30-s + 9·31-s − 21·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 9-s + 2.84·10-s − 5.19·12-s + 0.832·13-s − 2.40·14-s + 2.32·15-s + 15/4·16-s − 0.727·17-s − 2.12·18-s − 4.02·20-s − 1.96·21-s − 3.12·23-s + 6.12·24-s + 6/5·25-s − 1.76·26-s − 0.192·27-s + 3.40·28-s − 4.92·30-s + 1.61·31-s − 3.71·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 2 p T^{2} + 10 T^{3} + 2 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_g_k |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 4 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_g_e |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 24 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.11.a_j_y |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 40 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_g_bo |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 42 T^{2} + 84 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.d_bq_dg |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.19.a_j_ce |
| 23 | $S_4\times C_2$ | \( 1 + 15 T + 108 T^{2} + 24 p T^{3} + 108 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.p_ee_vg |
| 31 | $S_4\times C_2$ | \( 1 - 9 T + 108 T^{2} - 556 T^{3} + 108 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_ee_avk |
| 37 | $S_4\times C_2$ | \( 1 + 39 T^{2} - 232 T^{3} + 39 p T^{4} + p^{3} T^{6} \) | 3.37.a_bn_aiy |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 99 T^{2} - 528 T^{3} + 99 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.am_dv_aui |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 144 T^{2} + 758 T^{3} + 144 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.j_fo_bde |
| 47 | $S_4\times C_2$ | \( 1 + 45 T^{2} - 192 T^{3} + 45 p T^{4} + p^{3} T^{6} \) | 3.47.a_bt_ahk |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 468 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_da_asa |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 180 T^{2} + 1602 T^{3} + 180 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.p_gy_cjq |
| 61 | $S_4\times C_2$ | \( 1 + 15 T + 204 T^{2} + 1604 T^{3} + 204 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.p_hw_cjs |
| 67 | $S_4\times C_2$ | \( 1 - 18 T + 3 p T^{2} - 1676 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.as_ht_acmm |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) | 3.71.bk_yv_kdc |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 234 T^{2} - 1312 T^{3} + 234 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_ja_abym |
| 79 | $S_4\times C_2$ | \( 1 + 9 T + 102 T^{2} + 320 T^{3} + 102 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.j_dy_mi |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1920 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.am_jp_acvw |
| 89 | $S_4\times C_2$ | \( 1 + 243 T^{2} + 24 T^{3} + 243 p T^{4} + p^{3} T^{6} \) | 3.89.a_jj_y |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 282 T^{2} + 1748 T^{3} + 282 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.j_kw_cpg |
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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
−7.33261293264473763797590972551, −7.19735471935641763984499982815, −6.76554377369253095609826116379, −6.52184841289028865203766427569, −6.23705118908809275763399079619, −6.22529946615144882724929368319, −5.97463082779752667400406538140, −5.84914150183930731552020977151, −5.51906080922343085410202878610, −5.22236676929779954925035728845, −4.85607446087519613670094994869, −4.61231960258819431360741207287, −4.51261394633429110360590675937, −4.09710038128799872544068843545, −3.93676888993952847000053946737, −3.74691527248827546347167338167, −3.27017236266127823132748873125, −2.86577218263453043888696527131, −2.81197796589558462629517621189, −2.22005415866698655030290601348, −2.06105640091815656435718471663, −1.73488588598567100586818304300, −1.36102771400168423357987276228, −1.00267662260678339857422766117, −0.849408658229323528697586709086, 0, 0, 0,
0.849408658229323528697586709086, 1.00267662260678339857422766117, 1.36102771400168423357987276228, 1.73488588598567100586818304300, 2.06105640091815656435718471663, 2.22005415866698655030290601348, 2.81197796589558462629517621189, 2.86577218263453043888696527131, 3.27017236266127823132748873125, 3.74691527248827546347167338167, 3.93676888993952847000053946737, 4.09710038128799872544068843545, 4.51261394633429110360590675937, 4.61231960258819431360741207287, 4.85607446087519613670094994869, 5.22236676929779954925035728845, 5.51906080922343085410202878610, 5.84914150183930731552020977151, 5.97463082779752667400406538140, 6.22529946615144882724929368319, 6.23705118908809275763399079619, 6.52184841289028865203766427569, 6.76554377369253095609826116379, 7.19735471935641763984499982815, 7.33261293264473763797590972551
