| L(s) = 1 | + 2·2-s + 4-s − 2·8-s + 8·11-s − 4·16-s + 16·22-s + 16·23-s + 8·25-s − 8·29-s − 2·32-s − 8·37-s − 16·43-s + 8·44-s + 32·46-s + 16·50-s + 8·53-s − 16·58-s + 3·64-s + 24·67-s − 16·74-s + 32·79-s − 32·86-s − 16·88-s + 16·92-s + 8·100-s + 16·106-s − 8·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 2.41·11-s − 16-s + 3.41·22-s + 3.33·23-s + 8/5·25-s − 1.48·29-s − 0.353·32-s − 1.31·37-s − 2.43·43-s + 1.20·44-s + 4.71·46-s + 2.26·50-s + 1.09·53-s − 2.10·58-s + 3/8·64-s + 2.93·67-s − 1.85·74-s + 3.60·79-s − 3.45·86-s − 1.70·88-s + 1.66·92-s + 4/5·100-s + 1.55·106-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.341699970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.341699970\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ai_a_bn |
| 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ai_ba_aey_xv |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_q_a_pm |
| 17 | $C_2^3$ | \( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_q_a_abh |
| 19 | $C_2^3$ | \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_ag_a_amn |
| 23 | $C_2^2$ | \( ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.aq_fq_abnk_ikp |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.29.i_fk_bca_jog |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ack_a_egx |
| 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 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_abg_a_fje |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.43.q_ki_dlg_bdac |
| 47 | $C_2^3$ | \( 1 - 62 T^{2} + 1635 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ack_a_ckx |
| 53 | $C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.ai_acg_aey_mvz |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.59.a_k_a_afab |
| 61 | $C_2^3$ | \( 1 - 120 T^{2} + 10679 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_aeq_a_put |
| 67 | $C_2^2$ | \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ay_lm_afcy_bypn |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^3$ | \( 1 + 96 T^{2} + 3887 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_ds_a_ftn |
| 79 | $C_2^2$ | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abg_xm_amdc_euqt |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ki_a_buyo |
| 89 | $C_2^3$ | \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_aey_a_mnn |
| 97 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_lc_a_cgni |
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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
−7.10195216315226775515316139810, −6.78634092939243797916736960045, −6.73526373731699038662510680285, −6.62218823392098014037216335273, −6.53303832332096410301683178964, −6.15922629835446783543826966800, −5.83691878970931258526636267259, −5.43717091389875276575135105337, −5.34915736567075376994640027468, −5.08095414495171742943271294648, −4.94791757740060135942024572565, −4.72190497129070411274419810112, −4.64203502502904655967286817085, −3.89932083057343143078105657481, −3.84094846432175063535720605690, −3.78796800478002405974273529316, −3.44051663652896944893109986483, −3.35751949347075180898923514539, −2.76337635700734706949798087235, −2.71468604343279408383827100321, −2.25057248695087158925981099390, −1.60008844771083753488360983081, −1.57538423239937573055979176463, −0.968702465333726216086743931593, −0.64003733616123595950867641556,
0.64003733616123595950867641556, 0.968702465333726216086743931593, 1.57538423239937573055979176463, 1.60008844771083753488360983081, 2.25057248695087158925981099390, 2.71468604343279408383827100321, 2.76337635700734706949798087235, 3.35751949347075180898923514539, 3.44051663652896944893109986483, 3.78796800478002405974273529316, 3.84094846432175063535720605690, 3.89932083057343143078105657481, 4.64203502502904655967286817085, 4.72190497129070411274419810112, 4.94791757740060135942024572565, 5.08095414495171742943271294648, 5.34915736567075376994640027468, 5.43717091389875276575135105337, 5.83691878970931258526636267259, 6.15922629835446783543826966800, 6.53303832332096410301683178964, 6.62218823392098014037216335273, 6.73526373731699038662510680285, 6.78634092939243797916736960045, 7.10195216315226775515316139810
