| L(s) = 1 | + 6·7-s + 3·9-s + 6·17-s + 7·25-s − 12·31-s − 12·41-s + 18·47-s + 3·49-s + 18·63-s + 18·71-s − 40·73-s − 48·79-s − 7·81-s − 24·97-s + 36·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + 163-s + 167-s − 2·169-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 9-s + 1.45·17-s + 7/5·25-s − 2.15·31-s − 1.87·41-s + 2.62·47-s + 3/7·49-s + 2.26·63-s + 2.13·71-s − 4.68·73-s − 5.40·79-s − 7/9·81-s − 2.43·97-s + 3.30·119-s + 3.27·121-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 + 1.45·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105835601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105835601\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ad_a_q |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ah_a_y |
| 7 | $D_{4}$ | \( ( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ag_bh_aek_oe |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abk_a_vu |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ag_cv_ali_cvk |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ae_a_bby |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.23.a_do_a_esc |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_u_a_cqo |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.m_fo_bng_kly |
| 37 | $D_4\times C_2$ | \( 1 - 47 T^{2} + 1416 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abv_a_ccm |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.m_hc_cbc_rlm |
| 43 | $D_4\times C_2$ | \( 1 + 45 T^{2} + 3248 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_bt_a_euy |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.as_iz_adhe_baiy |
| 53 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aq_a_gs |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agi_a_ugk |
| 61 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 7854 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_cm_a_lqc |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aho_a_bbms |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.as_mr_affa_ccro |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.73.bo_bii_swu_hjrq |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.79.bw_btk_bbbk_lcag |
| 83 | $D_4\times C_2$ | \( 1 + 24 T^{2} - 1378 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_y_a_acba |
| 89 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_im_a_bpis |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.97.y_xg_lpw_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
−5.88881489367925788516322416605, −5.78999185020033052071437590762, −5.77456608081455986258947750684, −5.52325441733755042190066054958, −5.22575059220092961547577707033, −5.20166735043223570991607960739, −4.91181826225550846614341443688, −4.44834957778947609807247248706, −4.43622535351123300987322875267, −4.37735396619837845091908413562, −4.37238713255357445008592766817, −3.92213328531889158585304251438, −3.51954656648421544635650567804, −3.34292266594115109328160970438, −3.16912662417752424540438399517, −3.08643089963816314862126056757, −2.63751182541022459166877600932, −2.27204644541490561325791889677, −2.22140886891336591601816853840, −1.67369081986625876657116674392, −1.46467770073232427613820281232, −1.42695316621183409340099218446, −1.34045693703152084134240607509, −0.817506636328532558396630592378, −0.22050644469743581377487213713,
0.22050644469743581377487213713, 0.817506636328532558396630592378, 1.34045693703152084134240607509, 1.42695316621183409340099218446, 1.46467770073232427613820281232, 1.67369081986625876657116674392, 2.22140886891336591601816853840, 2.27204644541490561325791889677, 2.63751182541022459166877600932, 3.08643089963816314862126056757, 3.16912662417752424540438399517, 3.34292266594115109328160970438, 3.51954656648421544635650567804, 3.92213328531889158585304251438, 4.37238713255357445008592766817, 4.37735396619837845091908413562, 4.43622535351123300987322875267, 4.44834957778947609807247248706, 4.91181826225550846614341443688, 5.20166735043223570991607960739, 5.22575059220092961547577707033, 5.52325441733755042190066054958, 5.77456608081455986258947750684, 5.78999185020033052071437590762, 5.88881489367925788516322416605
