| L(s) = 1 | + 16·19-s − 4·25-s + 8·43-s − 2·49-s + 16·67-s + 40·73-s + 8·97-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.67·19-s − 4/5·25-s + 1.21·43-s − 2/7·49-s + 1.95·67-s + 4.68·73-s + 0.812·97-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.092776018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.092776018\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_e_a_cc |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abo_a_ys |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abs_a_bfq |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aca_a_bwg |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.19.aq_gq_absy_izm |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_dk_a_ele |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ei_a_hdi |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ca_a_fbi |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_adw_a_irm |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.43.ai_ho_aboy_tms |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_gq_a_rmk |
| 53 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ia_a_yic |
| 59 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ado_a_nle |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agq_a_vys |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.67.aq_oa_afdo_chgo |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_abo_a_pne |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) | 4.73.abo_bii_aswu_hjrq |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ajk_a_bomo |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aki_a_buyo |
| 89 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_adw_a_bbdm |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.97.ai_pw_admu_dmla |
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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
−6.01896351764703913675286487217, −5.66397755358475809196932631365, −5.51119939543041355124400338044, −5.48407526939148692751660258591, −5.02435063950794746196635316037, −4.91761113335368298621499944675, −4.89914018381804770602666375844, −4.86580638978457067379345072245, −4.41942171824294151749899064245, −3.86984562470340761029369274205, −3.81245103387978208764772076043, −3.75468363056921986166153969760, −3.66261365640882094465174384725, −3.42759838443524989188051426887, −2.92134837815372225782292882102, −2.88214537970609417047787286205, −2.69456685688658018429377411471, −2.36264499740899863886478637161, −2.08575011881872810591284746599, −1.91719057252140360797009512174, −1.56106498477646771898271987144, −1.04827222856337348425303151538, −0.961249061160140955981512298851, −0.885933188659881201346982323460, −0.27999804239279711271409905805,
0.27999804239279711271409905805, 0.885933188659881201346982323460, 0.961249061160140955981512298851, 1.04827222856337348425303151538, 1.56106498477646771898271987144, 1.91719057252140360797009512174, 2.08575011881872810591284746599, 2.36264499740899863886478637161, 2.69456685688658018429377411471, 2.88214537970609417047787286205, 2.92134837815372225782292882102, 3.42759838443524989188051426887, 3.66261365640882094465174384725, 3.75468363056921986166153969760, 3.81245103387978208764772076043, 3.86984562470340761029369274205, 4.41942171824294151749899064245, 4.86580638978457067379345072245, 4.89914018381804770602666375844, 4.91761113335368298621499944675, 5.02435063950794746196635316037, 5.48407526939148692751660258591, 5.51119939543041355124400338044, 5.66397755358475809196932631365, 6.01896351764703913675286487217
