| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·6-s + 6·7-s + 2·8-s + 9-s + 6·11-s + 2·12-s − 12·14-s − 4·16-s − 2·18-s + 6·19-s + 12·21-s − 12·22-s + 6·23-s + 4·24-s − 14·25-s − 2·27-s + 6·28-s + 6·29-s − 24·31-s + 2·32-s + 12·33-s + 36-s + 6·37-s − 12·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s + 2.26·7-s + 0.707·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 3.20·14-s − 16-s − 0.471·18-s + 1.37·19-s + 2.61·21-s − 2.55·22-s + 1.25·23-s + 0.816·24-s − 2.79·25-s − 0.384·27-s + 1.13·28-s + 1.11·29-s − 4.31·31-s + 0.353·32-s + 2.08·33-s + 1/6·36-s + 0.986·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.140457364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140457364\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 13 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_o_a_dv |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 16 T^{2} - 36 T^{3} + 99 T^{4} - 36 p T^{5} + 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ag_q_abk_dv |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ag_i_abk_kh |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ah_a_ajg |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_ai_abk_bih |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 108 T^{3} - 573 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ag_i_ee_awb |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ag_abf_acc_dwe |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.y_me_eea_bazu |
| 37 | $C_2^2$ | \( ( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ag_abv_acc_gfc |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 43 T^{2} + 18 T^{3} + 3084 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ag_abr_s_eoq |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 52 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.c_ace_aca_cit |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.am_jc_acpw_bais |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) | 4.53.am_kg_aczo_bhnn |
| 59 | $C_2^3$ | \( 1 + 74 T^{2} + 1995 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_cw_a_cyt |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_hx_cee_oqy |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 1404 p T^{5} + 112 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.as_ei_acca_bbtb |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 88 T^{2} + 108 T^{3} + 7779 T^{4} + 108 p T^{5} - 88 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ag_adk_ee_lnf |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ac_a_ptz |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.i_eu_boy_banm |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.am_jc_adgm_bsak |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 58 T^{2} - 288 T^{3} + 20067 T^{4} - 288 p T^{5} - 58 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.am_acg_alc_bdrv |
| 97 | $C_2^2$ | \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.am_adi_aqq_brrn |
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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
−7.44840132380951396286984624919, −7.22966169474984014512374531515, −6.75598840639314417028290807737, −6.58413143715596786274542641419, −6.48239898712289480976095206789, −5.78697512592380010045927215883, −5.65941499594437548263762881360, −5.64515965657425916307585467019, −5.41389751332613500273024066505, −5.14382556717242297272576842049, −4.61337771539561811427005324939, −4.47653124226387003591104945691, −4.47274577948414626852604407914, −3.87946565583440586531951309197, −3.75376449236174848968684612391, −3.59447505604814877523106573955, −3.47388006322466673863389024397, −2.87722101140396126600806197990, −2.31489505239269390877729029464, −2.20124469829245014992331495060, −2.14713353501892156329977930257, −1.41446469989974304690030244998, −1.40079287153463705573707305331, −1.15583859476880636239869723132, −0.49164293832461253000007305805,
0.49164293832461253000007305805, 1.15583859476880636239869723132, 1.40079287153463705573707305331, 1.41446469989974304690030244998, 2.14713353501892156329977930257, 2.20124469829245014992331495060, 2.31489505239269390877729029464, 2.87722101140396126600806197990, 3.47388006322466673863389024397, 3.59447505604814877523106573955, 3.75376449236174848968684612391, 3.87946565583440586531951309197, 4.47274577948414626852604407914, 4.47653124226387003591104945691, 4.61337771539561811427005324939, 5.14382556717242297272576842049, 5.41389751332613500273024066505, 5.64515965657425916307585467019, 5.65941499594437548263762881360, 5.78697512592380010045927215883, 6.48239898712289480976095206789, 6.58413143715596786274542641419, 6.75598840639314417028290807737, 7.22966169474984014512374531515, 7.44840132380951396286984624919
