| L(s) = 1 | + 2·2-s − 4-s + 8·5-s − 8·8-s + 6·9-s + 16·10-s − 7·16-s − 20·17-s + 12·18-s − 8·20-s + 20·25-s − 16·29-s + 14·32-s − 40·34-s − 6·36-s + 16·37-s − 64·40-s + 48·45-s + 14·49-s + 40·50-s − 32·58-s + 35·64-s + 20·68-s − 48·72-s + 36·73-s + 32·74-s − 56·80-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 3.57·5-s − 2.82·8-s + 2·9-s + 5.05·10-s − 7/4·16-s − 4.85·17-s + 2.82·18-s − 1.78·20-s + 4·25-s − 2.97·29-s + 2.47·32-s − 6.85·34-s − 36-s + 2.63·37-s − 10.1·40-s + 7.15·45-s + 2·49-s + 5.65·50-s − 4.20·58-s + 35/8·64-s + 2.42·68-s − 5.65·72-s + 4.21·73-s + 3.71·74-s − 6.26·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.346115439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.346115439\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.5.ai_bs_afw_pq |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ao_a_fr |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bm_a_xf |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_ak_a_nz |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.17.u_ik_cgm_lax |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_cs_a_cwx |
| 23 | $C_2^2$ | \( ( 1 - 39 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ada_a_dvf |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.29.q_ie_clk_qcc |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_dw_a_goc |
| 41 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_e_a_ezm |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_fs_a_now |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_abc_a_gvm |
| 53 | $C_2^2$ | \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_gk_a_snb |
| 59 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agy_a_whi |
| 61 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_acy_a_ndu |
| 67 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_abs_a_nzy |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ka_a_bnxu |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.73.abk_bdy_apzk_gfzf |
| 79 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_acq_a_ueo |
| 83 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ha_a_bgql |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.89.au_tm_aiqq_egsx |
| 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
−7.912140857833570302814699596337, −7.78731918289715332310538821877, −7.56118712355965080694290999631, −7.09807787899631153088354790038, −6.63090620050528318969280245861, −6.62535069334038516408617672862, −6.56751445120358751023074880841, −6.20207276909779878980190238011, −5.83958868373555651296820329044, −5.79905802730777312574778309199, −5.65667512615616551110606000397, −5.23417650169453857335688151663, −4.91165158456628831090492700243, −4.70300418337615310597113079434, −4.50041555626685332607700856597, −4.13545106937363147769783192069, −3.99161731176240275856749085230, −3.75950022471336226640861295919, −3.36631615861563643398851542967, −2.49711101138027854331800500116, −2.20457800785985166375679621968, −2.18063606413675979907030990401, −2.15715156734931586962107591531, −1.54480024289498613714572009359, −0.60551681926434948693777038682,
0.60551681926434948693777038682, 1.54480024289498613714572009359, 2.15715156734931586962107591531, 2.18063606413675979907030990401, 2.20457800785985166375679621968, 2.49711101138027854331800500116, 3.36631615861563643398851542967, 3.75950022471336226640861295919, 3.99161731176240275856749085230, 4.13545106937363147769783192069, 4.50041555626685332607700856597, 4.70300418337615310597113079434, 4.91165158456628831090492700243, 5.23417650169453857335688151663, 5.65667512615616551110606000397, 5.79905802730777312574778309199, 5.83958868373555651296820329044, 6.20207276909779878980190238011, 6.56751445120358751023074880841, 6.62535069334038516408617672862, 6.63090620050528318969280245861, 7.09807787899631153088354790038, 7.56118712355965080694290999631, 7.78731918289715332310538821877, 7.912140857833570302814699596337
