| L(s) = 1 | − 2·9-s + 8·11-s + 2·25-s + 8·29-s − 20·37-s − 16·43-s − 12·53-s − 24·67-s − 16·79-s + 9·81-s − 16·99-s + 24·107-s − 4·109-s + 40·113-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.41·11-s + 2/5·25-s + 1.48·29-s − 3.28·37-s − 2.43·43-s − 1.64·53-s − 2.93·67-s − 1.80·79-s + 81-s − 1.60·99-s + 2.32·107-s − 0.383·109-s + 3.76·113-s + 3.45·121-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 − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881605477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881605477\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_c_a_af |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) | 4.5.a_ac_a_av |
| 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ai_ba_aey_xv |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_bk_a_zm |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ac_a_akz |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_abe_a_ut |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.29.ai_fk_abca_jog |
| 31 | $C_2^3$ | \( 1 - 30 T^{2} - 61 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_abe_a_acj |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.37.u_is_cyy_uwp |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_dw_a_irm |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.43.q_ki_dlg_bdac |
| 47 | $C_2^3$ | \( 1 - 62 T^{2} + 1635 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ack_a_ckx |
| 53 | $C_2^2$ | \( ( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_c_qq_ojz |
| 59 | $C_2^3$ | \( 1 - 110 T^{2} + 8619 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aeg_a_mtn |
| 61 | $C_2^3$ | \( 1 + 78 T^{2} + 2363 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_da_a_dmx |
| 67 | $C_2^2$ | \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.y_lm_fcy_bypn |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_afq_a_xqx |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.q_bi_bnk_bhtr |
| 83 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_acq_a_wck |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_agw_a_bjdz |
| 97 | $C_2^2$ | \( ( 1 + 162 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_mm_a_core |
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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
−8.286556423169087707926098207668, −8.202594605936697197156932967220, −7.54927402229830459217100187465, −7.30849982546782090853070611163, −7.25791786273739748446173331153, −6.92691518206747670015331844446, −6.64948777162028041469149311143, −6.52904760707153411109532012994, −6.04540078146282509217719979313, −5.97880008697827446947481344152, −5.94513077887181066549754004669, −5.17735157149894786222184523191, −5.14522847595527537625941839725, −4.67003510677040555077644354434, −4.57349176867595145680435130491, −4.38631423690551057138172782844, −3.69040152908058084237100284160, −3.51881024026257597992203295995, −3.38239692407249307717048776738, −3.04604672505750898743805004954, −2.67285277110792870429230406627, −1.92158084688416263093246023122, −1.57578267648255255967579757385, −1.52621829710490321215170211141, −0.54052466127689685608185063784,
0.54052466127689685608185063784, 1.52621829710490321215170211141, 1.57578267648255255967579757385, 1.92158084688416263093246023122, 2.67285277110792870429230406627, 3.04604672505750898743805004954, 3.38239692407249307717048776738, 3.51881024026257597992203295995, 3.69040152908058084237100284160, 4.38631423690551057138172782844, 4.57349176867595145680435130491, 4.67003510677040555077644354434, 5.14522847595527537625941839725, 5.17735157149894786222184523191, 5.94513077887181066549754004669, 5.97880008697827446947481344152, 6.04540078146282509217719979313, 6.52904760707153411109532012994, 6.64948777162028041469149311143, 6.92691518206747670015331844446, 7.25791786273739748446173331153, 7.30849982546782090853070611163, 7.54927402229830459217100187465, 8.202594605936697197156932967220, 8.286556423169087707926098207668
