| L(s) = 1 | − 4·4-s + 6·7-s + 2·13-s + 12·16-s − 2·25-s − 24·28-s − 12·31-s − 4·37-s − 12·43-s + 7·49-s − 8·52-s − 22·61-s − 32·64-s + 42·67-s − 4·73-s + 6·79-s + 12·91-s + 26·97-s + 8·100-s − 30·103-s − 40·109-s + 72·112-s − 2·121-s + 48·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.26·7-s + 0.554·13-s + 3·16-s − 2/5·25-s − 4.53·28-s − 2.15·31-s − 0.657·37-s − 1.82·43-s + 49-s − 1.10·52-s − 2.81·61-s − 4·64-s + 5.13·67-s − 0.468·73-s + 0.675·79-s + 1.25·91-s + 2.63·97-s + 4/5·100-s − 2.95·103-s − 3.83·109-s + 6.80·112-s − 0.181·121-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211646154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211646154\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_c_a_av |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) | 4.7.ag_bd_ady_mm |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_c_a_aen |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ac_ax_ac_to |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aca_a_bwg |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) | 4.19.a_aw_a_bgl |
| 23 | $C_2^3$ | \( 1 - 22 T^{2} - 45 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_aw_a_abt |
| 29 | $C_2^3$ | \( 1 + 26 T^{2} - 165 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ba_a_agj |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.m_es_bie_iwx |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.37.e_fy_rg_mvb |
| 41 | $C_2^3$ | \( 1 + 50 T^{2} + 819 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_by_a_bfn |
| 43 | $C_2^2$ | \( ( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.m_fq_btg_onr |
| 47 | $C_2^3$ | \( 1 - 70 T^{2} + 2691 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_acs_a_dzn |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afs_a_qks |
| 59 | $C_2^3$ | \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_adq_a_hxz |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.w_jh_dyk_bmem |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \) | 4.67.abq_bhl_arly_gmlo |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.73.e_lm_bhw_bwpn |
| 79 | $C_2^2$ | \( ( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ag_gr_able_benk |
| 83 | $C_2^3$ | \( 1 - 70 T^{2} - 1989 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_acs_a_acyn |
| 89 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_anc_a_coew |
| 97 | $C_2^2$ | \( ( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.aba_mb_agna_dgae |
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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
−8.333831959649029851067196926577, −8.284311391516580194323789608815, −7.929258475368335618894650602707, −7.88503049448383049836368763587, −7.72721308057389181645785926366, −7.35364872910435467386172281747, −6.75033457302711834331528391974, −6.61886577976753218685881837302, −6.60539158364233464493275247096, −5.89657157271422704362469481788, −5.46929717547565204087265273560, −5.45311809145221515340641399476, −5.26394759092418200156097167610, −5.09912967013019240860000542272, −4.59254157472082617100481322493, −4.40533686011668830291725867325, −4.25396527917363217207197367179, −3.75136778458233557217615719392, −3.48440990481653988386812738108, −3.34942834674788102263952355676, −2.71937789360993006903007416664, −1.97650555028363986605070807297, −1.61397288815048488848598089754, −1.49632007829764188428102868808, −0.53330560875433874219015734496,
0.53330560875433874219015734496, 1.49632007829764188428102868808, 1.61397288815048488848598089754, 1.97650555028363986605070807297, 2.71937789360993006903007416664, 3.34942834674788102263952355676, 3.48440990481653988386812738108, 3.75136778458233557217615719392, 4.25396527917363217207197367179, 4.40533686011668830291725867325, 4.59254157472082617100481322493, 5.09912967013019240860000542272, 5.26394759092418200156097167610, 5.45311809145221515340641399476, 5.46929717547565204087265273560, 5.89657157271422704362469481788, 6.60539158364233464493275247096, 6.61886577976753218685881837302, 6.75033457302711834331528391974, 7.35364872910435467386172281747, 7.72721308057389181645785926366, 7.88503049448383049836368763587, 7.929258475368335618894650602707, 8.284311391516580194323789608815, 8.333831959649029851067196926577
