| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 3·9-s + 2·12-s − 14-s − 16-s − 3·18-s − 8·19-s − 2·21-s − 6·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s − 5·32-s − 3·36-s + 12·37-s + 8·38-s + 2·42-s + 2·48-s + 49-s + 6·50-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s + 0.577·12-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 1.83·19-s − 0.436·21-s − 1.22·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s − 0.883·32-s − 1/2·36-s + 1.97·37-s + 1.29·38-s + 0.308·42-s + 0.288·48-s + 1/7·49-s + 0.848·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.863214598051417092763914791206, −9.198895146278336601365818133221, −9.163045893798789171919999202805, −8.068098981215555715874277637796, −7.977686193886306753143677629185, −7.38325958034671995701748313853, −6.53484921474790582970192984243, −6.19643457342138342025418009798, −5.48510765590071063887274763802, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −4.01671863219499060051377207353, −2.39234840548789475685588947322, −1.40786771277750203137263322761, 0,
1.40786771277750203137263322761, 2.39234840548789475685588947322, 4.01671863219499060051377207353, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.48510765590071063887274763802, 6.19643457342138342025418009798, 6.53484921474790582970192984243, 7.38325958034671995701748313853, 7.977686193886306753143677629185, 8.068098981215555715874277637796, 9.163045893798789171919999202805, 9.198895146278336601365818133221, 9.863214598051417092763914791206
