| L(s) = 1 | + 8·13-s − 20·25-s − 2·49-s − 56·61-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 2.21·13-s − 4·25-s − 2/7·49-s − 7.17·61-s − 4·121-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 − 0.923·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242516485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242516485\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.5.a_u_a_fu |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.11.a_bs_a_bby |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.13.ai_cy_ang_clq |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ca_a_cbu |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.23.a_ado_a_esc |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ado_a_fzi |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.37.a_aca_a_fbi |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.59.a_ajc_a_bexi |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) | 4.61.ce_ccq_bfki_lqck |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_jk_a_bjhu |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.73.a_do_a_sxi |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aky_a_bwhq |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.89.a_ans_a_cshy |
| 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
−6.02531288589133181974785974644, −5.92037058083739490436620535020, −5.85786076859312040007306860622, −5.21772892173754901226165429290, −5.13990685550451191565453393425, −5.01932708388041304516580445284, −4.95170363464776228791418730624, −4.39454215780990071575395926195, −4.31632737346917584689058785434, −4.05489222235286947751049286689, −4.04823026715330309603184572432, −3.74379319773414750979862277512, −3.39144039885011359499924398264, −3.39070215828890222431854312023, −3.33328704947412590409297021598, −2.72586901540268924191710107701, −2.58968125582364013823318101683, −2.42252837745485319963731045865, −2.12582251292371223293178145075, −1.58708634058047786718257389312, −1.42327199973411291946639229461, −1.40623653113136167794615730567, −1.37282414110226615238196677767, −0.41307889716314904907502465453, −0.22199967097526857505873866772,
0.22199967097526857505873866772, 0.41307889716314904907502465453, 1.37282414110226615238196677767, 1.40623653113136167794615730567, 1.42327199973411291946639229461, 1.58708634058047786718257389312, 2.12582251292371223293178145075, 2.42252837745485319963731045865, 2.58968125582364013823318101683, 2.72586901540268924191710107701, 3.33328704947412590409297021598, 3.39070215828890222431854312023, 3.39144039885011359499924398264, 3.74379319773414750979862277512, 4.04823026715330309603184572432, 4.05489222235286947751049286689, 4.31632737346917584689058785434, 4.39454215780990071575395926195, 4.95170363464776228791418730624, 5.01932708388041304516580445284, 5.13990685550451191565453393425, 5.21772892173754901226165429290, 5.85786076859312040007306860622, 5.92037058083739490436620535020, 6.02531288589133181974785974644
