| L(s) = 1 | − 6·9-s + 4·13-s − 16·19-s − 8·31-s + 20·37-s − 16·43-s − 4·49-s + 28·61-s + 8·79-s + 27·81-s + 28·97-s + 40·103-s + 4·109-s − 24·117-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 96·171-s + 173-s + ⋯ |
| L(s) = 1 | − 2·9-s + 1.10·13-s − 3.67·19-s − 1.43·31-s + 3.28·37-s − 2.43·43-s − 4/7·49-s + 3.58·61-s + 0.900·79-s + 3·81-s + 2.84·97-s + 3.94·103-s + 0.383·109-s − 2.21·117-s − 4·121-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 + 8/13·169-s + 7.34·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7052120591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7052120591\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.5.a_a_a_by |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_e_a_dy |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.11.a_bs_a_bby |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 22 T^{2} + p^{2} T^{4} ) \) | 4.13.ae_i_bk_amk |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.17.a_a_a_wg |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) | 4.19.q_ey_bbs_ezu |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.23.a_a_a_bos |
| 29 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.29.a_a_a_cms |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) | 4.31.i_bg_aeq_acmc |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 22 T^{2} + p^{2} T^{4} ) \) | 4.43.q_ey_my_pi |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.47.a_ahg_a_tpu |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.59.a_a_a_khu |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 74 T^{2} + p^{2} T^{4} ) \) | 4.61.abc_pc_afpk_btvi |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_jk_a_bjhu |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ado_a_sxi |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 - 142 T^{2} + p^{2} T^{4} ) \) | 4.79.ai_bg_tk_ascg |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.89.a_a_a_xli |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) | 4.97.abc_pc_aecq_bczu |
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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
−8.121018374979836109623625924393, −7.84381486968434112143157597117, −7.80464200683867183147098572149, −7.42702986091449735857314617454, −6.83306912568438797047703823163, −6.65437099763126580942894701983, −6.53397041892081126124094409566, −6.25228262033485203933217489929, −6.13629005715920118926255702671, −5.97475483929843681934793661730, −5.43946976352792035550145054328, −5.35053099334061609816908453183, −5.03191405785265011562290557908, −4.62266596664040337141551898971, −4.50935003856798378118467835069, −3.97864743980852044947191244695, −3.74069099374504300200137465317, −3.55728559248602678271593426072, −3.35463673912773952847432490018, −2.55232849460477640372081045369, −2.42177172426919177307168989860, −2.34264823937798931907499410302, −1.81140915550003515510116415230, −1.12041197563818132412706438004, −0.31506473622181132043367193386,
0.31506473622181132043367193386, 1.12041197563818132412706438004, 1.81140915550003515510116415230, 2.34264823937798931907499410302, 2.42177172426919177307168989860, 2.55232849460477640372081045369, 3.35463673912773952847432490018, 3.55728559248602678271593426072, 3.74069099374504300200137465317, 3.97864743980852044947191244695, 4.50935003856798378118467835069, 4.62266596664040337141551898971, 5.03191405785265011562290557908, 5.35053099334061609816908453183, 5.43946976352792035550145054328, 5.97475483929843681934793661730, 6.13629005715920118926255702671, 6.25228262033485203933217489929, 6.53397041892081126124094409566, 6.65437099763126580942894701983, 6.83306912568438797047703823163, 7.42702986091449735857314617454, 7.80464200683867183147098572149, 7.84381486968434112143157597117, 8.121018374979836109623625924393
