| L(s) = 1 | + 4-s − 4·7-s − 2·13-s + 4·16-s + 8·19-s + 10·25-s − 4·28-s − 4·31-s + 8·37-s − 16·43-s + 18·49-s − 2·52-s + 20·61-s + 11·64-s − 28·67-s − 40·73-s + 8·76-s + 8·79-s + 8·91-s + 20·97-s + 10·100-s + 8·103-s − 40·109-s − 16·112-s + 10·121-s − 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 0.554·13-s + 16-s + 1.83·19-s + 2·25-s − 0.755·28-s − 0.718·31-s + 1.31·37-s − 2.43·43-s + 18/7·49-s − 0.277·52-s + 2.56·61-s + 11/8·64-s − 3.42·67-s − 4.68·73-s + 0.917·76-s + 0.900·79-s + 0.838·91-s + 2.03·97-s + 100-s + 0.788·103-s − 3.83·109-s − 1.51·112-s + 0.909·121-s − 0.359·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206607104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206607104\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_ab_a_ad |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ak_a_cx |
| 7 | $C_2^2$ | \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.e_ac_q_gh |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ak_a_av |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abc_a_bdu |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.19.ai_dw_asu_epa |
| 23 | $C_2^3$ | \( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_c_a_auf |
| 29 | $C_2^3$ | \( 1 - 10 T^{2} - 741 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ak_a_abcn |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.e_aby_q_ehn |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.37.ai_gq_abjk_ouw |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abi_a_auf |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) | 4.43.q_ec_bnk_ogx |
| 47 | $C_2^3$ | \( 1 + 14 T^{2} - 2013 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_o_a_aczl |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.53.a_ie_a_yyg |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aec_a_lmh |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.au_gw_acyy_bfhz |
| 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 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ka_a_bnxu |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.73.bo_bii_swu_hjrq |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) | 4.79.ai_aeg_aey_bcbz |
| 83 | $C_2^3$ | \( 1 - 58 T^{2} - 3525 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_acg_a_affp |
| 89 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ka_a_bwli |
| 97 | $C_2^2$ | \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.au_ec_acyy_ceoh |
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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
−7.16383027361883716861421423649, −6.81212599430114716819481596758, −6.79266204156901055362942714101, −6.44566810446066994817835562061, −6.19913530600869018581305081790, −5.97692031537625201361094346871, −5.79219159593553451082544357016, −5.55531485142104464988690298879, −5.23111765031276849190284604383, −5.19583723703184579293876340547, −4.84507190469352194562572237324, −4.48259994347699662815431413041, −4.42181702344029603547187654249, −3.92555349760032697002934419721, −3.60936104244745954265496485949, −3.48665465979800042378836803340, −3.20790951191979285032723499507, −3.02844547110334376926926724354, −2.67970593258259612319484830021, −2.38072212126293163029197944523, −2.36793952614415448231409789973, −1.38525139558497184314692634632, −1.34449922770271394112339018246, −1.07819062623861571055832585069, −0.25336292695304605514231517887,
0.25336292695304605514231517887, 1.07819062623861571055832585069, 1.34449922770271394112339018246, 1.38525139558497184314692634632, 2.36793952614415448231409789973, 2.38072212126293163029197944523, 2.67970593258259612319484830021, 3.02844547110334376926926724354, 3.20790951191979285032723499507, 3.48665465979800042378836803340, 3.60936104244745954265496485949, 3.92555349760032697002934419721, 4.42181702344029603547187654249, 4.48259994347699662815431413041, 4.84507190469352194562572237324, 5.19583723703184579293876340547, 5.23111765031276849190284604383, 5.55531485142104464988690298879, 5.79219159593553451082544357016, 5.97692031537625201361094346871, 6.19913530600869018581305081790, 6.44566810446066994817835562061, 6.79266204156901055362942714101, 6.81212599430114716819481596758, 7.16383027361883716861421423649
