| L(s) = 1 | + 2·3-s − 3·5-s − 4·7-s − 9-s + 10·11-s + 2·13-s − 6·15-s + 4·17-s + 10·19-s − 8·21-s − 4·23-s + 6·25-s − 4·27-s + 3·29-s + 2·31-s + 20·33-s + 12·35-s + 4·39-s − 10·41-s + 10·43-s + 3·45-s + 2·47-s − 5·49-s + 8·51-s − 6·53-s − 30·55-s + 20·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 1.51·7-s − 1/3·9-s + 3.01·11-s + 0.554·13-s − 1.54·15-s + 0.970·17-s + 2.29·19-s − 1.74·21-s − 0.834·23-s + 6/5·25-s − 0.769·27-s + 0.557·29-s + 0.359·31-s + 3.48·33-s + 2.02·35-s + 0.640·39-s − 1.56·41-s + 1.52·43-s + 0.447·45-s + 0.291·47-s − 5/7·49-s + 1.12·51-s − 0.824·53-s − 4.04·55-s + 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.438026881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.438026881\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_f_ai |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 52 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_v_ca |
| 11 | $S_4\times C_2$ | \( 1 - 10 T + 57 T^{2} - 216 T^{3} + 57 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ak_cf_aii |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bb_abs |
| 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 | $S_4\times C_2$ | \( 1 - 10 T + 85 T^{2} - 400 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ak_dh_apk |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 68 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_bp_cq |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 17 T^{2} - 240 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ac_r_ajg |
| 37 | $S_4\times C_2$ | \( 1 + 83 T^{2} + 52 T^{3} + 83 p T^{4} + p^{3} T^{6} \) | 3.37.a_df_ca |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 284 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_ct_ky |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 157 T^{2} - 880 T^{3} + 157 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_gb_abhw |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 304 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ac_cn_als |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 155 T^{2} + 596 T^{3} + 155 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_fz_wy |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 81 T^{2} - 8 p T^{3} + 81 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.am_dd_ase |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} + 28 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ag_ch_bc |
| 67 | $S_4\times C_2$ | \( 1 + 101 T^{2} - 4 p T^{3} + 101 p T^{4} + p^{3} T^{6} \) | 3.67.a_dx_aki |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 181 T^{2} + 864 T^{3} + 181 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_gz_bhg |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 259 T^{2} - 2236 T^{3} + 259 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aq_jz_adia |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 429 T^{2} + 4544 T^{3} + 429 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ba_qn_gsu |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 209 T^{2} - 596 T^{3} + 209 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ae_ib_awy |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 167 T^{2} + 1964 T^{3} + 167 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.k_gl_cxo |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 331 T^{2} + 3004 T^{3} + 331 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.q_mt_elo |
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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
−6.78835416491012015205794701224, −6.77141811505159077917448552450, −6.32025067282221488502259906806, −6.26212341147975202652411873531, −5.92873744174826461241282541558, −5.70726032848242226601734775158, −5.61294946942053258007316869684, −5.07467948954224550797628879412, −4.92276770761238353920567446782, −4.67179736690665691714473790576, −4.10460388261397054987069961803, −4.07848111345202008632575182034, −3.95256763191548579967467996682, −3.52423735666598517625664710510, −3.43644685173533365160823041317, −3.36959319473122244691899656089, −2.90324809330195729668322538961, −2.81667713620122622081487791382, −2.70361210840117308421469978661, −1.93411283615617140838692286801, −1.71141802265181148738543491877, −1.33230264197718407778162272489, −1.10718241403911205678162646117, −0.61325679278779131570352500133, −0.46920500650092001468318091304,
0.46920500650092001468318091304, 0.61325679278779131570352500133, 1.10718241403911205678162646117, 1.33230264197718407778162272489, 1.71141802265181148738543491877, 1.93411283615617140838692286801, 2.70361210840117308421469978661, 2.81667713620122622081487791382, 2.90324809330195729668322538961, 3.36959319473122244691899656089, 3.43644685173533365160823041317, 3.52423735666598517625664710510, 3.95256763191548579967467996682, 4.07848111345202008632575182034, 4.10460388261397054987069961803, 4.67179736690665691714473790576, 4.92276770761238353920567446782, 5.07467948954224550797628879412, 5.61294946942053258007316869684, 5.70726032848242226601734775158, 5.92873744174826461241282541558, 6.26212341147975202652411873531, 6.32025067282221488502259906806, 6.77141811505159077917448552450, 6.78835416491012015205794701224
