L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 8·11-s + 12-s − 4·13-s − 16-s + 18-s − 8·22-s + 3·24-s − 6·25-s − 4·26-s − 27-s + 5·32-s + 8·33-s − 36-s + 12·37-s + 4·39-s + 8·44-s + 48-s + 49-s − 6·50-s + 4·52-s − 54-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.235·18-s − 1.70·22-s + 0.612·24-s − 6/5·25-s − 0.784·26-s − 0.192·27-s + 0.883·32-s + 1.39·33-s − 1/6·36-s + 1.97·37-s + 0.640·39-s + 1.20·44-s + 0.144·48-s + 1/7·49-s − 0.848·50-s + 0.554·52-s − 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−10.51953132809604428367636932730, −10.18331996404873608095625884352, −9.457705101580370517739336150901, −9.198895146278336601365818133221, −8.068098981215555715874277637796, −7.81295430367747747098376616892, −7.38325958034671995701748313853, −6.19643457342138342025418009798, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −3.05422074105458389777226041971, −2.39234840548789475685588947322, 0,
2.39234840548789475685588947322, 3.05422074105458389777226041971, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 6.19643457342138342025418009798, 7.38325958034671995701748313853, 7.81295430367747747098376616892, 8.068098981215555715874277637796, 9.198895146278336601365818133221, 9.457705101580370517739336150901, 10.18331996404873608095625884352, 10.51953132809604428367636932730