L(s) = 1 | + 4·2-s + 12·4-s + 32·8-s − 80·11-s + 80·16-s − 320·22-s − 136·23-s − 200·25-s − 220·29-s + 192·32-s − 40·37-s − 680·43-s − 960·44-s − 544·46-s − 800·50-s − 1.25e3·53-s − 880·58-s + 448·64-s + 1.08e3·67-s + 840·71-s − 160·74-s − 1.52e3·79-s − 2.72e3·86-s − 2.56e3·88-s − 1.63e3·92-s − 2.40e3·100-s − 5.02e3·106-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2.19·11-s + 5/4·16-s − 3.10·22-s − 1.23·23-s − 8/5·25-s − 1.40·29-s + 1.06·32-s − 0.177·37-s − 2.41·43-s − 3.28·44-s − 1.74·46-s − 2.26·50-s − 3.25·53-s − 1.99·58-s + 7/8·64-s + 1.96·67-s + 1.40·71-s − 0.251·74-s − 2.16·79-s − 3.41·86-s − 3.10·88-s − 1.84·92-s − 2.39·100-s − 4.60·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 8 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 40 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 344 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9824 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13590 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 68 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 45470 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 135392 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 340 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 199454 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 628 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 358042 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 388440 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 693984 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 760 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 251126 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 81488 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1573296 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.725778701660704215345246628442, −9.344291069557412162470974285445, −8.315745238565526962526080568310, −8.134114786299807859530940129162, −7.79553157910202937552323822878, −7.50938534127597541854384262774, −6.66998073637732415002969678633, −6.55567738434376934896898272451, −5.80289680844946262702665755637, −5.51534287221672004091105183355, −5.12902023974388648388445275842, −4.79739450911468622155658371538, −3.90243426658539376455343675128, −3.85198287833411627621161539920, −3.01618959758702576407865535448, −2.67601088857169002793436145572, −1.90353166781167779560196388647, −1.66156170822587452417560838736, 0, 0,
1.66156170822587452417560838736, 1.90353166781167779560196388647, 2.67601088857169002793436145572, 3.01618959758702576407865535448, 3.85198287833411627621161539920, 3.90243426658539376455343675128, 4.79739450911468622155658371538, 5.12902023974388648388445275842, 5.51534287221672004091105183355, 5.80289680844946262702665755637, 6.55567738434376934896898272451, 6.66998073637732415002969678633, 7.50938534127597541854384262774, 7.79553157910202937552323822878, 8.134114786299807859530940129162, 8.315745238565526962526080568310, 9.344291069557412162470974285445, 9.725778701660704215345246628442