| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 9-s − 2·10-s + 2·12-s − 15-s − 4·16-s + 2·18-s − 2·20-s + 12·23-s + 25-s + 27-s + 10·29-s − 2·30-s − 8·32-s + 2·36-s + 8·43-s − 45-s + 24·46-s + 4·47-s − 4·48-s − 10·49-s + 2·50-s − 8·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s − 0.258·15-s − 16-s + 0.471·18-s − 0.447·20-s + 2.50·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.365·30-s − 1.41·32-s + 1/3·36-s + 1.21·43-s − 0.149·45-s + 3.53·46-s + 0.583·47-s − 0.577·48-s − 1.42·49-s + 0.282·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.162069031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.162069031\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.963134976875332410550837868755, −8.657968102763697321620128690150, −7.998362829294470925327343124130, −7.53885251381440898433031798219, −7.01325793013184944469112654608, −6.50757219156439507333081149228, −6.21841312997577567139299736727, −5.35930529872856387675389024109, −4.87191825277757691038211661892, −4.61465994793805005819745900336, −3.96187051825471841097070685011, −3.20581269290493402875849813307, −3.01888100048798853558195810431, −2.28832826282711051669350915460, −1.06272008942114379011094572988,
1.06272008942114379011094572988, 2.28832826282711051669350915460, 3.01888100048798853558195810431, 3.20581269290493402875849813307, 3.96187051825471841097070685011, 4.61465994793805005819745900336, 4.87191825277757691038211661892, 5.35930529872856387675389024109, 6.21841312997577567139299736727, 6.50757219156439507333081149228, 7.01325793013184944469112654608, 7.53885251381440898433031798219, 7.998362829294470925327343124130, 8.657968102763697321620128690150, 8.963134976875332410550837868755
