| L(s) = 1 | + 6·3-s − 5·4-s + 27·9-s − 30·12-s − 26·13-s + 9·16-s − 2·25-s + 108·27-s − 135·36-s − 156·39-s − 140·43-s + 54·48-s + 98·49-s + 130·52-s + 140·61-s + 35·64-s − 12·75-s − 100·79-s + 405·81-s + 10·100-s + 100·103-s − 540·108-s − 702·117-s + 190·121-s + 127-s − 840·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2·3-s − 5/4·4-s + 3·9-s − 5/2·12-s − 2·13-s + 9/16·16-s − 0.0799·25-s + 4·27-s − 3.75·36-s − 4·39-s − 3.25·43-s + 9/8·48-s + 2·49-s + 5/2·52-s + 2.29·61-s + 0.546·64-s − 0.159·75-s − 1.26·79-s + 5·81-s + 1/10·100-s + 0.970·103-s − 5·108-s − 6·117-s + 1.57·121-s + 0.00787·127-s − 6.51·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.486855218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486855218\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 190 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2930 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4370 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13730 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9550 T^{2} + p^{4} T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−16.17776076440684107365168140773, −15.41296427192426339951561739008, −14.74779851860642345067934614351, −14.71888502141935727087379748289, −13.91684005007903017929791009855, −13.55702950455440272715149582818, −12.98958346172328036324827187800, −12.47565412749268450916641274269, −11.73597428151827597838532486081, −10.29328181329658191819376378800, −9.844698820623542068247718320755, −9.529095596346153519991494946556, −8.598115240214397110623217254118, −8.419010208358207271646465835522, −7.42024942755176890131207590194, −6.94756332994272473301724931275, −5.10598999469327975101040915001, −4.46224924269202224452794575203, −3.50592891452292509657612261607, −2.31693734272072535421877652429,
2.31693734272072535421877652429, 3.50592891452292509657612261607, 4.46224924269202224452794575203, 5.10598999469327975101040915001, 6.94756332994272473301724931275, 7.42024942755176890131207590194, 8.419010208358207271646465835522, 8.598115240214397110623217254118, 9.529095596346153519991494946556, 9.844698820623542068247718320755, 10.29328181329658191819376378800, 11.73597428151827597838532486081, 12.47565412749268450916641274269, 12.98958346172328036324827187800, 13.55702950455440272715149582818, 13.91684005007903017929791009855, 14.71888502141935727087379748289, 14.74779851860642345067934614351, 15.41296427192426339951561739008, 16.17776076440684107365168140773
