| L(s) = 1 | − 2·4-s + 3·9-s − 16·11-s + 3·16-s + 2·19-s − 2·29-s − 18·31-s − 6·36-s − 16·41-s + 32·44-s − 8·49-s + 26·59-s + 38·61-s − 4·64-s − 42·71-s − 4·76-s − 7·81-s + 14·89-s − 48·99-s + 8·101-s + 26·109-s + 4·116-s + 116·121-s + 36·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 9-s − 4.82·11-s + 3/4·16-s + 0.458·19-s − 0.371·29-s − 3.23·31-s − 36-s − 2.49·41-s + 4.82·44-s − 8/7·49-s + 3.38·59-s + 4.86·61-s − 1/2·64-s − 4.98·71-s − 0.458·76-s − 7/9·81-s + 1.48·89-s − 4.82·99-s + 0.796·101-s + 2.49·109-s + 0.371·116-s + 10.5·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7817797488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7817797488\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 5 | | \( 1 \) | |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ad_a_q |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 46 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_i_a_bu |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.11.q_fk_bee_epa |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_abf_a_se |
| 19 | $D_{4}$ | \( ( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ac_cr_aec_cvs |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ace_a_cqg |
| 29 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.c_bp_du_dei |
| 31 | $D_{4}$ | \( ( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.s_jd_cxm_thc |
| 37 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4430 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_ads_a_gok |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.q_eu_brs_obu |
| 43 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6686 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_aeq_a_jxe |
| 47 | $D_4\times C_2$ | \( 1 - 139 T^{2} + 8904 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_afj_a_nem |
| 53 | $D_4\times C_2$ | \( 1 - 199 T^{2} + 15480 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ahr_a_wxk |
| 59 | $D_{4}$ | \( ( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.aba_px_agza_cjvg |
| 61 | $D_{4}$ | \( ( 1 - 19 T + 208 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.abm_bdx_apde_fkee |
| 67 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15742 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ahc_a_xhm |
| 71 | $D_{4}$ | \( ( 1 + 21 T + 214 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bq_bhl_rsk_gths |
| 73 | $D_4\times C_2$ | \( 1 - 55 T^{2} + 6208 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_acd_a_jeu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 21526 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ahg_a_bfvy |
| 89 | $D_{4}$ | \( ( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ao_np_aezu_csnk |
| 97 | $D_4\times C_2$ | \( 1 - 271 T^{2} + 34080 T^{4} - 271 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_akl_a_byku |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29140338375142982622832240796, −7.14296398412646785149910700538, −7.12237449955511959619547328927, −6.74929042748835889138657864642, −6.46954481546696752075061937615, −5.91221709301221156041849206098, −5.66923789787085704032810633207, −5.59465057718043016443589477415, −5.44983853280781921941344891556, −5.15649114311591850240191736452, −4.98259556072740577699873537870, −4.92737034035962650804129147810, −4.64533597781767906258719247013, −4.04045539871305934225924780676, −4.02132983269878194245048342304, −3.58575278116087150754053511930, −3.49689508302728237672250331629, −3.01503767476057869584094690851, −2.73061490539498997647281389437, −2.62233734761671294920664206920, −1.96410154343783385734264831723, −1.95809256434979639500389585690, −1.55752144630233094404833744687, −0.48501491581702052714904256773, −0.42247611161915083596435617575,
0.42247611161915083596435617575, 0.48501491581702052714904256773, 1.55752144630233094404833744687, 1.95809256434979639500389585690, 1.96410154343783385734264831723, 2.62233734761671294920664206920, 2.73061490539498997647281389437, 3.01503767476057869584094690851, 3.49689508302728237672250331629, 3.58575278116087150754053511930, 4.02132983269878194245048342304, 4.04045539871305934225924780676, 4.64533597781767906258719247013, 4.92737034035962650804129147810, 4.98259556072740577699873537870, 5.15649114311591850240191736452, 5.44983853280781921941344891556, 5.59465057718043016443589477415, 5.66923789787085704032810633207, 5.91221709301221156041849206098, 6.46954481546696752075061937615, 6.74929042748835889138657864642, 7.12237449955511959619547328927, 7.14296398412646785149910700538, 7.29140338375142982622832240796
