| L(s) = 1 | + 4-s + 4·5-s − 2·7-s − 3·9-s + 8·13-s + 4·20-s + 14·25-s − 2·28-s − 4·29-s − 8·35-s − 3·36-s − 12·45-s + 15·49-s + 8·52-s + 8·53-s − 8·59-s + 6·63-s − 64-s + 32·65-s − 4·67-s + 28·71-s + 8·83-s − 16·91-s + 14·100-s − 38·103-s + 16·107-s + 24·109-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 0.755·7-s − 9-s + 2.21·13-s + 0.894·20-s + 14/5·25-s − 0.377·28-s − 0.742·29-s − 1.35·35-s − 1/2·36-s − 1.78·45-s + 15/7·49-s + 1.10·52-s + 1.09·53-s − 1.04·59-s + 0.755·63-s − 1/8·64-s + 3.96·65-s − 0.488·67-s + 3.32·71-s + 0.878·83-s − 1.67·91-s + 7/5·100-s − 3.74·103-s + 1.54·107-s + 2.29·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.603231577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.603231577\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ae_c_aq_dn |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.7.c_al_c_fs |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_v_a_mi |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ai_w_aey_bdj |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.17.a_aci_a_cew |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acw_a_dcl |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) | 4.31.a_bu_a_bsl |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.a_afk_a_lhu |
| 41 | $C_2^3$ | \( 1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_da_a_gnj |
| 43 | $C_2^3$ | \( 1 + 5 T^{2} - 1824 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_f_a_acse |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_cg_a_bsl |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.53.ai_jc_abye_bcss |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.i_acs_ey_pvn |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) | 4.61.a_abv_a_acge |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.e_aes_q_typ |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) | 4.71.abc_wg_akwe_egbj |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aka_a_boty |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) | 4.79.a_fm_a_upn |
| 83 | $C_2^2$ | \( ( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ai_aeo_aey_beyt |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ank_a_cqfu |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.97.a_fa_a_lcd |
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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
−7.941316809998093987961059294570, −7.73117960773942752842724811904, −7.21630099609704423589416707605, −6.93538508916802539225330471461, −6.69273677992369047720059006923, −6.65054285836619692060347953310, −6.49185508220658472555346951209, −6.10761774798077706922814241669, −5.74726152148926872082731622915, −5.72975903247426715770396270385, −5.55683410122019095833534661071, −5.32709319370395485649534850855, −5.04312037863447606746042840224, −4.49826042255930150332052968818, −4.14393261632435862506790836279, −4.07191293077392805058901120471, −3.61512115301385519379151004856, −3.07717395764334057481192608604, −3.01540017284380410530336775221, −3.01070681632793610055022701291, −2.11094735221923997629919290130, −2.10047390762500387938419578849, −1.83182998458002102256063989657, −0.993592364786548688367774994223, −0.823847506012596813373372613877,
0.823847506012596813373372613877, 0.993592364786548688367774994223, 1.83182998458002102256063989657, 2.10047390762500387938419578849, 2.11094735221923997629919290130, 3.01070681632793610055022701291, 3.01540017284380410530336775221, 3.07717395764334057481192608604, 3.61512115301385519379151004856, 4.07191293077392805058901120471, 4.14393261632435862506790836279, 4.49826042255930150332052968818, 5.04312037863447606746042840224, 5.32709319370395485649534850855, 5.55683410122019095833534661071, 5.72975903247426715770396270385, 5.74726152148926872082731622915, 6.10761774798077706922814241669, 6.49185508220658472555346951209, 6.65054285836619692060347953310, 6.69273677992369047720059006923, 6.93538508916802539225330471461, 7.21630099609704423589416707605, 7.73117960773942752842724811904, 7.941316809998093987961059294570
