| L(s) = 1 | − 10·25-s + 36·41-s + 2·49-s + 32·73-s − 48·89-s + 4·97-s + 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2·25-s + 5.62·41-s + 2/7·49-s + 3.74·73-s − 5.08·89-s + 0.406·97-s + 1.12·113-s + 2.36·121-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 + 1.69·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.669091197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669091197\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_k_a_cx |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ac_a_dv |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aba_a_pv |
| 13 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_aw_a_rr |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.17.a_cq_a_cos |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ck_a_czr |
| 29 | $C_2^2$ | \( ( 1 - 43 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adi_a_ffv |
| 31 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_dq_a_gcx |
| 37 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_abc_a_eiw |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.41.abk_za_akwm_dfpb |
| 43 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_acw_a_hmx |
| 47 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_gc_a_ptz |
| 53 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ado_a_llm |
| 59 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aik_a_bbwt |
| 61 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ba_a_lgt |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aba_a_nnv |
| 71 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_gi_a_ywk |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.73.abg_baa_anki_fghq |
| 79 | $C_2^2$ | \( ( 1 + 143 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_la_a_bwsp |
| 83 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_amc_a_cevz |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.89.bw_buy_bdeu_mqmo |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.97.ae_pe_absy_dhgd |
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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
−5.68793326382312851355371499749, −5.55929686803401364751719266920, −5.52492556381354424861781055472, −5.49215187953267746303535660750, −4.99877125473703777248970250041, −4.64211820488592452604053629191, −4.52033187249174691141340100053, −4.49228644557331374837173766173, −4.14573474010062086206016142710, −4.09897782843367829277407432733, −3.96587038302867859220209648448, −3.61222971903952200281175538646, −3.42882407918160190508487138136, −3.06987678743561230661554204782, −2.96487516212965279805701137204, −2.83751655588806434616108523381, −2.39537292370687170905673875542, −2.19682465392589172178157380561, −2.10795434901207246918686510072, −1.85830905057666613296032775862, −1.62959433602507900189961485757, −0.954766117641276624792135858477, −0.874259496250278819102729167889, −0.816968258340082756882501170372, −0.21984827749133720401209097180,
0.21984827749133720401209097180, 0.816968258340082756882501170372, 0.874259496250278819102729167889, 0.954766117641276624792135858477, 1.62959433602507900189961485757, 1.85830905057666613296032775862, 2.10795434901207246918686510072, 2.19682465392589172178157380561, 2.39537292370687170905673875542, 2.83751655588806434616108523381, 2.96487516212965279805701137204, 3.06987678743561230661554204782, 3.42882407918160190508487138136, 3.61222971903952200281175538646, 3.96587038302867859220209648448, 4.09897782843367829277407432733, 4.14573474010062086206016142710, 4.49228644557331374837173766173, 4.52033187249174691141340100053, 4.64211820488592452604053629191, 4.99877125473703777248970250041, 5.49215187953267746303535660750, 5.52492556381354424861781055472, 5.55929686803401364751719266920, 5.68793326382312851355371499749
