| L(s) = 1 | + 4-s − 5-s + 9-s + 16-s − 8·19-s − 20-s + 25-s − 12·29-s + 16·31-s + 36-s − 12·41-s − 45-s + 2·49-s − 20·61-s + 64-s − 8·76-s + 16·79-s − 80-s + 81-s + 36·89-s + 8·95-s + 100-s + 36·101-s − 20·109-s − 12·116-s − 22·121-s + 16·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.447·5-s + 1/3·9-s + 1/4·16-s − 1.83·19-s − 0.223·20-s + 1/5·25-s − 2.22·29-s + 2.87·31-s + 1/6·36-s − 1.87·41-s − 0.149·45-s + 2/7·49-s − 2.56·61-s + 1/8·64-s − 0.917·76-s + 1.80·79-s − 0.111·80-s + 1/9·81-s + 3.81·89-s + 0.820·95-s + 1/10·100-s + 3.58·101-s − 1.91·109-s − 1.11·116-s − 2·121-s + 1.43·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8374489837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8374489837\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−12.14360527021257208948458691105, −11.91033651235536617108337917161, −11.23148855774643011734788423771, −10.40288135541038540368452071068, −10.39160509435207624542212960871, −9.305587122869086271308905422277, −8.749862280172923617009160642326, −7.941231322257043910223245152113, −7.56689807773085737112376729195, −6.42217617652666865799421983967, −6.38949936725272530282824625186, −5.07079211702149946545201877286, −4.30912662284802770612117339996, −3.36585804145949552210873743484, −2.07452981833800877942993394416,
2.07452981833800877942993394416, 3.36585804145949552210873743484, 4.30912662284802770612117339996, 5.07079211702149946545201877286, 6.38949936725272530282824625186, 6.42217617652666865799421983967, 7.56689807773085737112376729195, 7.941231322257043910223245152113, 8.749862280172923617009160642326, 9.305587122869086271308905422277, 10.39160509435207624542212960871, 10.40288135541038540368452071068, 11.23148855774643011734788423771, 11.91033651235536617108337917161, 12.14360527021257208948458691105
