| L(s) = 1 | − 6·9-s + 20·25-s + 4·29-s + 20·41-s − 28·49-s − 28·53-s + 27·81-s − 44·89-s + 52·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·9-s + 4·25-s + 0.742·29-s + 3.12·41-s − 4·49-s − 3.84·53-s + 3·81-s − 4.66·89-s + 4.89·113-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 + 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346652936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346652936\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^3$ | \( 1 + 14 T^{4} + p^{4} T^{8} \) | 4.11.a_a_a_o |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 23 | $C_2^3$ | \( 1 - 994 T^{4} + p^{4} T^{8} \) | 4.23.a_a_a_abmg |
| 29 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ae_i_aeu_cvu |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_bc_a_ddm |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.au_hs_acsa_uuw |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^3$ | \( 1 - 1282 T^{4} + p^{4} T^{8} \) | 4.47.a_a_a_abxi |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bc_pc_ggq_cbgo |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aie_a_bbqk |
| 67 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aem_a_sgs |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_agi_a_zso |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_alg_a_bxzy |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_dwg |
| 89 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bs_blg_voe_jdni |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_a_dfni |
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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
−6.62449514319353327625309911287, −6.37696693546886761206020012309, −6.12364534792701644258551561639, −6.11327913765840005013945121844, −5.70285726557789157108830183917, −5.48586010784546322832984297182, −5.40417028026792763349865992143, −5.03837537705732389762607420446, −4.83539306118379441392692677569, −4.59984072291268201704721571731, −4.58915606077743505266272497061, −4.30299222891199411618513131576, −4.05409858181632121658076833489, −3.48450686236720634745427680225, −3.23172985773847861645871169499, −3.15753719801612509456920747833, −2.91389590819140860697041989697, −2.87001296722371171137090158985, −2.65810292285209026249132951813, −2.12819357288588547944061904219, −1.79193742586027092202425268943, −1.52971410339891823029183867507, −1.09829149167705271207331536228, −0.69536154569393828303743325714, −0.35313356086883437621426677474,
0.35313356086883437621426677474, 0.69536154569393828303743325714, 1.09829149167705271207331536228, 1.52971410339891823029183867507, 1.79193742586027092202425268943, 2.12819357288588547944061904219, 2.65810292285209026249132951813, 2.87001296722371171137090158985, 2.91389590819140860697041989697, 3.15753719801612509456920747833, 3.23172985773847861645871169499, 3.48450686236720634745427680225, 4.05409858181632121658076833489, 4.30299222891199411618513131576, 4.58915606077743505266272497061, 4.59984072291268201704721571731, 4.83539306118379441392692677569, 5.03837537705732389762607420446, 5.40417028026792763349865992143, 5.48586010784546322832984297182, 5.70285726557789157108830183917, 6.11327913765840005013945121844, 6.12364534792701644258551561639, 6.37696693546886761206020012309, 6.62449514319353327625309911287
