| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 12-s − 2·14-s + 16-s − 18-s + 10·19-s + 2·21-s − 24-s + 2·25-s + 27-s + 2·28-s − 8·31-s − 32-s + 36-s − 8·37-s − 10·38-s − 2·42-s + 48-s − 3·49-s − 2·50-s − 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.534·14-s + 1/4·16-s − 0.235·18-s + 2.29·19-s + 0.436·21-s − 0.204·24-s + 2/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s − 0.176·32-s + 1/6·36-s − 1.31·37-s − 1.62·38-s − 0.308·42-s + 0.144·48-s − 3/7·49-s − 0.282·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328659576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328659576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
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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
−10.05672027087538034197729084707, −9.669446228193820242625181572817, −9.191843170597370710676249873197, −8.710988309948881821027132572543, −8.151975308258651318483216887612, −7.71357947029418561689741506363, −7.17775054949742134187241067553, −6.83722363424200252735152593821, −5.83930380628444089133080655376, −5.26719013957172234419382032551, −4.76207130767025177348761542541, −3.63831478244584951417136526212, −3.20883331189918105671695816383, −2.14994026496101327134244552530, −1.27768042596702657001936796215,
1.27768042596702657001936796215, 2.14994026496101327134244552530, 3.20883331189918105671695816383, 3.63831478244584951417136526212, 4.76207130767025177348761542541, 5.26719013957172234419382032551, 5.83930380628444089133080655376, 6.83722363424200252735152593821, 7.17775054949742134187241067553, 7.71357947029418561689741506363, 8.151975308258651318483216887612, 8.710988309948881821027132572543, 9.191843170597370710676249873197, 9.669446228193820242625181572817, 10.05672027087538034197729084707
