| L(s) = 1 | + 8·11-s + 4·25-s − 24·43-s − 2·49-s − 40·67-s + 40·107-s + 80·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 4/5·25-s − 3.65·43-s − 2/7·49-s − 4.88·67-s + 3.86·107-s + 7.52·113-s − 0.363·121-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 − 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570212347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570212347\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ae_a_cc |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.11.ai_cq_alk_bww |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_bc_a_uo |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abs_a_bow |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_u_a_bfq |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_adg_a_eeo |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_m_a_coc |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_dw_a_goc |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aem_a_jas |
| 41 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_ca_a_fzi |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.43.y_oy_fwi_btva |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_do_a_jri |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.53.a_agy_a_uhq |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_afk_a_rog |
| 61 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_bc_a_lhu |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.67.bo_bhk_rvc_grow |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_akq_a_brcg |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_abc_a_stq |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ajc_a_bozm |
| 89 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_afk_a_beru |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.97.a_ae_a_bbvy |
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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
−6.52999818297702145770484102771, −6.46396352608426803712122327761, −6.08815385016969837811744283929, −5.90238963513722074058434871722, −5.87041481421635675175517501127, −5.41887222563925915534616406117, −5.16635735401514488869056422593, −4.98539896647011394067067904365, −4.75904160110613806107040159345, −4.48564073533462642338115497971, −4.47312255326184185596777631776, −4.15554221876154850017801763133, −3.89688921835590284302222761554, −3.49772147805126382808465517191, −3.42071198001946732271832097405, −3.27841452061145411424953372485, −2.96777773718522621503093443680, −2.82473363265895859427487140283, −2.19191925955932652414462408680, −2.00922021995486447538183576224, −1.76150318082949363647688594827, −1.46057021991227438632198589917, −1.18594378711053803493586101894, −0.907434723999026460602655249734, −0.21143321977326660974870902711,
0.21143321977326660974870902711, 0.907434723999026460602655249734, 1.18594378711053803493586101894, 1.46057021991227438632198589917, 1.76150318082949363647688594827, 2.00922021995486447538183576224, 2.19191925955932652414462408680, 2.82473363265895859427487140283, 2.96777773718522621503093443680, 3.27841452061145411424953372485, 3.42071198001946732271832097405, 3.49772147805126382808465517191, 3.89688921835590284302222761554, 4.15554221876154850017801763133, 4.47312255326184185596777631776, 4.48564073533462642338115497971, 4.75904160110613806107040159345, 4.98539896647011394067067904365, 5.16635735401514488869056422593, 5.41887222563925915534616406117, 5.87041481421635675175517501127, 5.90238963513722074058434871722, 6.08815385016969837811744283929, 6.46396352608426803712122327761, 6.52999818297702145770484102771
