| L(s) = 1 | + 4-s + 2·7-s + 24·19-s − 14·25-s + 2·28-s − 12·31-s − 8·37-s + 16·43-s − 11·49-s − 12·61-s − 64-s − 28·67-s − 42·73-s + 24·76-s − 22·79-s + 24·97-s − 14·100-s − 32·109-s + 44·121-s − 12·124-s + 127-s + 131-s + 48·133-s + 137-s + 139-s − 8·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 5.50·19-s − 2.79·25-s + 0.377·28-s − 2.15·31-s − 1.31·37-s + 2.43·43-s − 1.57·49-s − 1.53·61-s − 1/8·64-s − 3.42·67-s − 4.91·73-s + 2.75·76-s − 2.47·79-s + 2.43·97-s − 7/5·100-s − 3.06·109-s + 4·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261443352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261443352\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_o_a_dv |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.11.a_abs_a_bby |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_ba_a_tn |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aw_a_hn |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ay_ks_adbk_pux |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_au_a_bso |
| 29 | $C_2^3$ | \( 1 - 23 T^{2} - 312 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ax_a_ama |
| 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 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.i_aba_ey_glv |
| 41 | $C_2^3$ | \( 1 - 70 T^{2} + 3219 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acs_a_etv |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.43.aq_ec_abnk_ogx |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ade_a_grr |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_dt_a_jtw |
| 59 | $C_2^3$ | \( 1 + 29 T^{2} - 2640 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_bd_a_adxo |
| 61 | $C_2^2$ | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.m_ha_cjw_zkd |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.bc_rm_idc_cytn |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aie_a_bfny |
| 73 | $C_2^2$ | \( ( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.bq_bhx_sfi_hapw |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.w_hx_dyk_bxjg |
| 83 | $C_2^3$ | \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_fe_a_qjr |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_acs_a_aemf |
| 97 | $C_2^2$ | \( ( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ay_qs_aipk_dwgx |
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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
−7.32499937904793829968906574658, −7.16179633061442411212625238381, −6.43833051108593193571865909884, −6.26323293432823735284398229035, −6.05093445852663352246468487451, −5.86800043356852777970644962712, −5.62489608776219843822928289588, −5.48816446748388282156307967318, −5.41943662621103522210635532679, −4.87914742664241811987270976646, −4.80682614768122444897990905927, −4.72256886851633731956872472458, −4.10475765593241965676458394856, −3.97429049954081141146203355572, −3.70296973349955313367848209947, −3.46626130918610480432785189005, −3.13952738172133391689860311797, −2.95700016599323887989685097185, −2.69575563735575496139580642784, −2.47879456273766357897552518537, −1.62266904897607752513575859145, −1.61655782865770239630900231396, −1.46923222089454465193784103114, −1.20161706674668295012847074045, −0.21845658474372100863496275516,
0.21845658474372100863496275516, 1.20161706674668295012847074045, 1.46923222089454465193784103114, 1.61655782865770239630900231396, 1.62266904897607752513575859145, 2.47879456273766357897552518537, 2.69575563735575496139580642784, 2.95700016599323887989685097185, 3.13952738172133391689860311797, 3.46626130918610480432785189005, 3.70296973349955313367848209947, 3.97429049954081141146203355572, 4.10475765593241965676458394856, 4.72256886851633731956872472458, 4.80682614768122444897990905927, 4.87914742664241811987270976646, 5.41943662621103522210635532679, 5.48816446748388282156307967318, 5.62489608776219843822928289588, 5.86800043356852777970644962712, 6.05093445852663352246468487451, 6.26323293432823735284398229035, 6.43833051108593193571865909884, 7.16179633061442411212625238381, 7.32499937904793829968906574658
