| L(s) = 1 | − 4·3-s − 8·5-s + 8·9-s − 12·11-s + 32·15-s − 4·23-s + 38·25-s − 20·27-s + 8·31-s + 48·33-s − 20·37-s − 64·45-s − 20·47-s − 36·53-s + 96·55-s − 20·67-s + 16·69-s − 152·75-s + 50·81-s − 32·93-s + 36·97-s − 96·99-s − 28·103-s + 80·111-s − 28·113-s + 32·115-s + 86·121-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.57·5-s + 8/3·9-s − 3.61·11-s + 8.26·15-s − 0.834·23-s + 38/5·25-s − 3.84·27-s + 1.43·31-s + 8.35·33-s − 3.28·37-s − 9.54·45-s − 2.91·47-s − 4.94·53-s + 12.9·55-s − 2.44·67-s + 1.92·69-s − 17.5·75-s + 50/9·81-s − 3.31·93-s + 3.65·97-s − 9.64·99-s − 2.75·103-s + 7.59·111-s − 2.63·113-s + 2.98·115-s + 7.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.e_i_u_bu |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_adq |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) | 4.13.a_a_a_fq |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) | 4.17.a_a_a_awc |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ci_a_ckk |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.e_i_dw_bvy |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dw_a_gew |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.31.ai_fs_abdw_ktq |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.u_hs_coy_ssc |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_au_a_fde |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) | 4.43.a_a_a_bug |
| 47 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.u_hs_cwq_ydq |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.53.bk_yy_lls_dtyk |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agi_a_ugk |
| 61 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aiu_a_begc |
| 67 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.u_hs_dma_bkuw |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) | 4.73.a_a_a_obq |
| 79 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_adg_a_vby |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_aikk |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abk_yy_anuq_gdqs |
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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
−7.65619063589752710484372446971, −7.49382275464458807772892580895, −7.38649459038551859949561035930, −7.12092400023061824768230959973, −7.00690589436325938162273676333, −6.34757464839445099400371132040, −6.33414693499900817645136912821, −6.16818490177671748339673456766, −6.14523056890622793546233690184, −5.37486055238421371680296680057, −5.16222297061185120263033971345, −5.15228116634536058765702853452, −5.10591139018875560091846792262, −4.76628025471263843135561319667, −4.55748530472669234876679429497, −4.34379521650161470135267128624, −3.91422207823689136597777433781, −3.72464596518172983768075891230, −3.30924330140769360707557078124, −3.20429266044975300391416457592, −2.97213189646694201226231967215, −2.64845879317920237954980199398, −1.99605417169297234664595941187, −1.63666087735133476902723427493, −1.10207585510823134476492867222, 0, 0, 0, 0,
1.10207585510823134476492867222, 1.63666087735133476902723427493, 1.99605417169297234664595941187, 2.64845879317920237954980199398, 2.97213189646694201226231967215, 3.20429266044975300391416457592, 3.30924330140769360707557078124, 3.72464596518172983768075891230, 3.91422207823689136597777433781, 4.34379521650161470135267128624, 4.55748530472669234876679429497, 4.76628025471263843135561319667, 5.10591139018875560091846792262, 5.15228116634536058765702853452, 5.16222297061185120263033971345, 5.37486055238421371680296680057, 6.14523056890622793546233690184, 6.16818490177671748339673456766, 6.33414693499900817645136912821, 6.34757464839445099400371132040, 7.00690589436325938162273676333, 7.12092400023061824768230959973, 7.38649459038551859949561035930, 7.49382275464458807772892580895, 7.65619063589752710484372446971
