L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2·11-s + 12-s − 8·13-s + 16-s − 18-s + 2·22-s + 12·23-s − 24-s − 10·25-s + 8·26-s + 27-s − 32-s − 2·33-s + 36-s − 20·37-s − 8·39-s − 2·44-s − 12·46-s − 12·47-s + 48-s − 10·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 2.21·13-s + 1/4·16-s − 0.235·18-s + 0.426·22-s + 2.50·23-s − 0.204·24-s − 2·25-s + 1.56·26-s + 0.192·27-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 3.28·37-s − 1.28·39-s − 0.301·44-s − 1.76·46-s − 1.75·47-s + 0.144·48-s − 1.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.245228842316309160133095693960, −8.958855021658912251315559727512, −8.129852226719742477393784673308, −8.013104010237496671597514230187, −7.33998580522115612678977642043, −6.79963832425739493066244038791, −6.77506565559004391547494912024, −5.43952609375822489573657963774, −5.21184114914749280771382385734, −4.69333471332706230755932945396, −3.64282367595579751815074401037, −3.08131485219503756183708188012, −2.37545091357255938865951638025, −1.70794665150414314375514308074, 0,
1.70794665150414314375514308074, 2.37545091357255938865951638025, 3.08131485219503756183708188012, 3.64282367595579751815074401037, 4.69333471332706230755932945396, 5.21184114914749280771382385734, 5.43952609375822489573657963774, 6.77506565559004391547494912024, 6.79963832425739493066244038791, 7.33998580522115612678977642043, 8.013104010237496671597514230187, 8.129852226719742477393784673308, 8.958855021658912251315559727512, 9.245228842316309160133095693960