| 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
| \(L(1)\) |
\(\approx\) |
\(0.8629900169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8629900169\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−12.67762165595594022358609556562, −12.21586614434777996929376757948, −11.63411923882571964461014443098, −11.06971333200683418097771407434, −10.61313903657403428631986258214, −9.838914541061391267445737913687, −9.162107724971553168279860855665, −8.209268299440378903633193192701, −7.67309113544266320982632690290, −6.75854972278424423989171001695, −6.12813525368893364682615715022, −5.29191319642104527806134908865, −4.83316120360162427340191670592, −3.58957782156774118206070509403, −2.48301597500780626280936465889,
2.48301597500780626280936465889, 3.58957782156774118206070509403, 4.83316120360162427340191670592, 5.29191319642104527806134908865, 6.12813525368893364682615715022, 6.75854972278424423989171001695, 7.67309113544266320982632690290, 8.209268299440378903633193192701, 9.162107724971553168279860855665, 9.838914541061391267445737913687, 10.61313903657403428631986258214, 11.06971333200683418097771407434, 11.63411923882571964461014443098, 12.21586614434777996929376757948, 12.67762165595594022358609556562
