| L(s) = 1 | + 4·13-s + 4·25-s + 8·37-s + 28·49-s + 8·73-s + 24·97-s − 24·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 4/5·25-s + 1.31·37-s + 4·49-s + 0.936·73-s + 2.43·97-s − 2.29·109-s − 3.63·121-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 + 0.769·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.605662750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605662750\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ae_a_cc |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.7.a_abc_a_li |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bo_a_ys |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bg_a_bgc |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_aca_a_cos |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aq_a_cpe |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_cy_a_ezm |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.37.ai_gq_abjk_ouw |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_abs_a_gew |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_ai_a_goo |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ei_a_mys |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_iy_a_befi |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_u_a_nle |
| 71 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_hc_a_bbli |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.73.ai_me_acqq_cane |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ahg_a_bfny |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_fg_a_bbfu |
| 89 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_bs_a_ydy |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.97.ay_xg_alpw_frkw |
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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
−5.47117222198854347786910101866, −5.45271592807476579267760538538, −5.28337234640885547172922471248, −4.99873725941054274761466132449, −4.70577480660332741525548669525, −4.57630548187118030362836190887, −4.48700019225958238902666577075, −4.11522937681562511896919911825, −4.06779425153178057092466838791, −3.80131596916930126289434329035, −3.75907362060574355474323511897, −3.42060691571976523990980114997, −3.31967960541853389510325929658, −2.97231228561854940824988167982, −2.70902513392419466041470977137, −2.70336592444895837173074253228, −2.32779880677057905290486739614, −2.13561968695398832967980268333, −2.09772607412952772399233940700, −1.59678063071626605709927457845, −1.33495150016840484747825556316, −1.04470537034628728411802261713, −0.875537529235343032917952258527, −0.78420530947993561690874146957, −0.13705019093600586929288962493,
0.13705019093600586929288962493, 0.78420530947993561690874146957, 0.875537529235343032917952258527, 1.04470537034628728411802261713, 1.33495150016840484747825556316, 1.59678063071626605709927457845, 2.09772607412952772399233940700, 2.13561968695398832967980268333, 2.32779880677057905290486739614, 2.70336592444895837173074253228, 2.70902513392419466041470977137, 2.97231228561854940824988167982, 3.31967960541853389510325929658, 3.42060691571976523990980114997, 3.75907362060574355474323511897, 3.80131596916930126289434329035, 4.06779425153178057092466838791, 4.11522937681562511896919911825, 4.48700019225958238902666577075, 4.57630548187118030362836190887, 4.70577480660332741525548669525, 4.99873725941054274761466132449, 5.28337234640885547172922471248, 5.45271592807476579267760538538, 5.47117222198854347786910101866
