L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 2·11-s + 12-s − 2·13-s + 16-s + 2·18-s − 2·22-s − 3·23-s − 24-s − 4·25-s + 2·26-s − 5·27-s − 32-s + 2·33-s − 2·36-s + 37-s − 2·39-s + 2·44-s + 3·46-s + 9·47-s + 48-s + 5·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.176·32-s + 0.348·33-s − 1/3·36-s + 0.164·37-s − 0.320·39-s + 0.301·44-s + 0.442·46-s + 1.31·47-s + 0.144·48-s + 5/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6250213940\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6250213940\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−12.65418471580462255718354903878, −11.94945510831953546666205870104, −11.67859104995133782855162915724, −10.89157814259766502301721731237, −10.22551646350884739634709025716, −9.617946753058921074165390870758, −8.993615420670583592832545430956, −8.579257706904438873046700044755, −7.70668237752242463432576097104, −7.33142598722630380941995415168, −6.24329912599518429624292681724, −5.68464252582251042101571546693, −4.40114328389798423754560516310, −3.32354453455858658608472519573, −2.18090952847020073511516588570,
2.18090952847020073511516588570, 3.32354453455858658608472519573, 4.40114328389798423754560516310, 5.68464252582251042101571546693, 6.24329912599518429624292681724, 7.33142598722630380941995415168, 7.70668237752242463432576097104, 8.579257706904438873046700044755, 8.993615420670583592832545430956, 9.617946753058921074165390870758, 10.22551646350884739634709025716, 10.89157814259766502301721731237, 11.67859104995133782855162915724, 11.94945510831953546666205870104, 12.65418471580462255718354903878