| L(s) = 1 | − 2·3-s + 3·9-s + 18·11-s + 4·19-s + 10·25-s − 10·27-s − 36·33-s + 18·41-s + 10·43-s − 14·49-s − 8·57-s − 18·59-s − 14·67-s − 4·73-s − 20·75-s + 20·81-s − 10·97-s + 54·99-s + 169·121-s − 36·123-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 5.42·11-s + 0.917·19-s + 2·25-s − 1.92·27-s − 6.26·33-s + 2.81·41-s + 1.52·43-s − 2·49-s − 1.05·57-s − 2.34·59-s − 1.71·67-s − 0.468·73-s − 2.30·75-s + 20/9·81-s − 1.01·97-s + 5.42·99-s + 15.3·121-s − 3.24·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236607894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236607894\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ak_a_cx |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_o_a_fr |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) | 4.11.as_fz_abgo_ewa |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_ba_a_tn |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) | 4.17.a_c_a_akz |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ae_aba_aq_bqh |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_acg_a_dtb |
| 31 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ck_a_egx |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) | 4.41.as_hd_acbi_mvs |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) | 4.43.ak_br_le_aeho |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_adq_a_jux |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.53.a_ie_a_yyg |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) | 4.59.s_hv_cnu_rqi |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_es_a_qnj |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} ) \) | 4.67.o_cp_acs_abls |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.e_afe_q_xrn |
| 79 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_gc_a_bbsd |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.83.a_gc_a_batf |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) | 4.89.a_alg_a_ccze |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.97.k_dt_acvm_abcgq |
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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
−8.827924567889045885301099642545, −8.501300124998138230446948964577, −7.920531447084458532137297802157, −7.76863507082012229687281988011, −7.55329826700543950434768949724, −7.10091141452713149675408339067, −6.83099914402236818248649814665, −6.78457114150948124790182012393, −6.63982992344918068078870872196, −6.01120858957617329698265157798, −6.00223255077593968430739697067, −5.97829348138928179141918463939, −5.64910693417385723551842872037, −4.79838545457740215565684306812, −4.79068267244108707273614832376, −4.48742206388738385160313811509, −4.18584635841734405353329323843, −3.87570584615837684313082337805, −3.68849337934032410300149581862, −3.26201932961832851480031272467, −2.90739748907543855475989015453, −2.14798322928933584975022696282, −1.44872290482539340360860925926, −1.24575039547965681227910336117, −1.06162082961248363009660896400,
1.06162082961248363009660896400, 1.24575039547965681227910336117, 1.44872290482539340360860925926, 2.14798322928933584975022696282, 2.90739748907543855475989015453, 3.26201932961832851480031272467, 3.68849337934032410300149581862, 3.87570584615837684313082337805, 4.18584635841734405353329323843, 4.48742206388738385160313811509, 4.79068267244108707273614832376, 4.79838545457740215565684306812, 5.64910693417385723551842872037, 5.97829348138928179141918463939, 6.00223255077593968430739697067, 6.01120858957617329698265157798, 6.63982992344918068078870872196, 6.78457114150948124790182012393, 6.83099914402236818248649814665, 7.10091141452713149675408339067, 7.55329826700543950434768949724, 7.76863507082012229687281988011, 7.920531447084458532137297802157, 8.501300124998138230446948964577, 8.827924567889045885301099642545
