| L(s) = 1 | + 4·5-s − 16·13-s + 4·17-s + 10·25-s − 4·29-s + 12·37-s − 32·41-s − 16·49-s − 32·53-s + 8·61-s − 64·65-s − 28·73-s − 18·81-s + 16·85-s − 24·89-s + 20·97-s − 64·109-s + 4·113-s + 8·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 4.43·13-s + 0.970·17-s + 2·25-s − 0.742·29-s + 1.97·37-s − 4.99·41-s − 2.28·49-s − 4.39·53-s + 1.02·61-s − 7.93·65-s − 3.27·73-s − 2·81-s + 1.73·85-s − 2.54·89-s + 2.03·97-s − 6.13·109-s + 0.376·113-s + 8/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.3.a_a_a_s |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_q_a_gg |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ai_a_jy |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.13.q_fs_bhw_fow |
| 17 | $D_{4}$ | \( ( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ae_ae_aca_bcc |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_cm_a_cpe |
| 23 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bo_a_cec |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_q_a_cyk |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.am_eu_ablk_kwg |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.41.bg_vc_iwe_cpoo |
| 43 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1778 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_abw_a_cqk |
| 47 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 2178 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abg_a_dfu |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.53.bg_wy_kom_dnfi |
| 59 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 2790 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_cy_a_edi |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ai_abs_ajo_pfy |
| 67 | $D_4\times C_2$ | \( 1 + 168 T^{2} + 13538 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_gm_a_uas |
| 71 | $D_4\times C_2$ | \( 1 + 220 T^{2} + 21558 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_im_a_bfxe |
| 73 | $D_{4}$ | \( ( 1 + 14 T + 156 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.bc_to_jmq_dqco |
| 79 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 3794 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_cu_a_fpy |
| 83 | $D_4\times C_2$ | \( 1 + 16 T^{2} - 8622 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_q_a_amtq |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.89.y_wa_kts_ezco |
| 97 | $D_{4}$ | \( ( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.au_rs_aifc_eamc |
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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
−5.78460607980125743837322375709, −5.43785902509247637496193684723, −5.23671872361453220071357871069, −5.22707525050213482509854208040, −5.14606834340862512713141474996, −4.93823382378972749944849841835, −4.81292706774008372298469450681, −4.55605527845847676352580795954, −4.35195192511055903020623842050, −4.32016443962882764092418546688, −4.06848380951836897064740162181, −3.63648029710399836810768819900, −3.34447822064616511235442712098, −3.10120494285328238284503419014, −3.09688820622047054888853903590, −3.06628149956187306110057488576, −2.67525479405873706509349313835, −2.49212629589033149269200043787, −2.27764878538914105480884237245, −2.18933375454083950557048453403, −1.74329496382749283583844026545, −1.64705491439172457898935022416, −1.48584929125612267540995026267, −1.30334960515876550207010559625, −0.992115198879370387654978850270, 0, 0, 0, 0,
0.992115198879370387654978850270, 1.30334960515876550207010559625, 1.48584929125612267540995026267, 1.64705491439172457898935022416, 1.74329496382749283583844026545, 2.18933375454083950557048453403, 2.27764878538914105480884237245, 2.49212629589033149269200043787, 2.67525479405873706509349313835, 3.06628149956187306110057488576, 3.09688820622047054888853903590, 3.10120494285328238284503419014, 3.34447822064616511235442712098, 3.63648029710399836810768819900, 4.06848380951836897064740162181, 4.32016443962882764092418546688, 4.35195192511055903020623842050, 4.55605527845847676352580795954, 4.81292706774008372298469450681, 4.93823382378972749944849841835, 5.14606834340862512713141474996, 5.22707525050213482509854208040, 5.23671872361453220071357871069, 5.43785902509247637496193684723, 5.78460607980125743837322375709
