| L(s) = 1 | + 4·5-s − 6·13-s − 16·17-s + 11·25-s − 4·29-s + 4·37-s + 16·41-s + 14·49-s − 16·53-s − 10·61-s − 24·65-s − 12·73-s − 64·85-s − 32·89-s + 36·97-s + 40·101-s + 12·109-s + 16·113-s + 22·121-s + 28·125-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.66·13-s − 3.88·17-s + 11/5·25-s − 0.742·29-s + 0.657·37-s + 2.49·41-s + 2·49-s − 2.19·53-s − 1.28·61-s − 2.97·65-s − 1.40·73-s − 6.94·85-s − 3.39·89-s + 3.65·97-s + 3.98·101-s + 1.14·109-s + 1.50·113-s + 2·121-s + 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750673334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750673334\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) | 4.5.ae_f_ae_q |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ao_a_fr |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aw_a_nz |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) | 4.13.g_n_as_aee |
| 17 | $C_2^2$ | \( ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.q_gc_bnk_hix |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.19.a_cy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) | 4.29.e_bd_aky_abrs |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ack_a_egx |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_ack_aq_gcp |
| 41 | $C_2^2$ | \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.aq_eg_abnk_nnn |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_adi_a_ifj |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_adq_a_jux |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.53.q_lw_eds_bojm |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aeo_a_plr |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.61.k_cj_abfy_amhg |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_afe_a_txz |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.m_abm_qq_zot |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_agc_a_bbsd |
| 83 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_agk_a_beox |
| 89 | $C_2^2$ | \( ( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bg_ws_mdc_fccp |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abk_bdy_argq_hpbf |
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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
−6.17131598576147665210803968424, −6.12511024314716596194618031746, −6.00186697089470922747680442243, −5.77688159139883416898236950329, −5.54799508037633283865008283116, −5.27889313501678102265872241344, −4.99525111679079482846731394702, −4.74159924378132057868790421317, −4.61027654244040808926046705488, −4.57937427688622491642882071640, −4.23470991645069879708163952933, −4.14207004787733053828312316234, −3.98010544142208454154197734933, −3.22908293305832255919377670398, −3.16856800590572243765022281199, −3.01073130148301822043233718155, −2.76953162033645582529117556317, −2.22869822164705421015879387963, −2.20394028021165459529155335462, −2.17574757971389090369059240755, −1.79904868561959792384250398213, −1.74961031628506591881867834168, −0.911550917391254516397338482955, −0.795281354231454063194466972172, −0.21473210720472761115649899956,
0.21473210720472761115649899956, 0.795281354231454063194466972172, 0.911550917391254516397338482955, 1.74961031628506591881867834168, 1.79904868561959792384250398213, 2.17574757971389090369059240755, 2.20394028021165459529155335462, 2.22869822164705421015879387963, 2.76953162033645582529117556317, 3.01073130148301822043233718155, 3.16856800590572243765022281199, 3.22908293305832255919377670398, 3.98010544142208454154197734933, 4.14207004787733053828312316234, 4.23470991645069879708163952933, 4.57937427688622491642882071640, 4.61027654244040808926046705488, 4.74159924378132057868790421317, 4.99525111679079482846731394702, 5.27889313501678102265872241344, 5.54799508037633283865008283116, 5.77688159139883416898236950329, 6.00186697089470922747680442243, 6.12511024314716596194618031746, 6.17131598576147665210803968424
