| L(s) = 1 | + 9-s − 2·29-s − 2·41-s + 49-s − 2·61-s + 81-s − 2·89-s + 2·101-s + 2·109-s + ⋯ |
| L(s) = 1 | + 9-s − 2·29-s − 2·41-s + 49-s − 2·61-s + 81-s − 2·89-s + 2·101-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8675353414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8675353414\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48043063637012269842053276802, −10.53102295877535198637202032984, −9.731389464909575187862516867306, −8.831791598922767076659082176671, −7.66643786576588532936556743744, −6.92206432134148200849355629544, −5.73724809922629184190843536622, −4.58535676496487356104408050937, −3.48309942335606106920691212917, −1.79991771110557346494545877197,
1.79991771110557346494545877197, 3.48309942335606106920691212917, 4.58535676496487356104408050937, 5.73724809922629184190843536622, 6.92206432134148200849355629544, 7.66643786576588532936556743744, 8.831791598922767076659082176671, 9.731389464909575187862516867306, 10.53102295877535198637202032984, 11.48043063637012269842053276802
