| L(s) = 1 | − 1.87·2-s + 2.53·4-s + 5-s − 2.87·8-s − 1.87·10-s + 2.87·16-s + 0.347·17-s + 0.347·19-s + 2.53·20-s + 1.53·23-s + 25-s − 1.87·31-s − 2.53·32-s − 0.652·34-s − 0.652·38-s − 2.87·40-s − 2.87·46-s − 47-s + 49-s − 1.87·50-s + 1.53·53-s − 1.87·61-s + 3.53·62-s + 1.87·64-s + 0.879·68-s + 0.879·76-s + 1.53·79-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.53·4-s + 5-s − 2.87·8-s − 1.87·10-s + 2.87·16-s + 0.347·17-s + 0.347·19-s + 2.53·20-s + 1.53·23-s + 25-s − 1.87·31-s − 2.53·32-s − 0.652·34-s − 0.652·38-s − 2.87·40-s − 2.87·46-s − 47-s + 49-s − 1.87·50-s + 1.53·53-s − 1.87·61-s + 3.53·62-s + 1.87·64-s + 0.879·68-s + 0.879·76-s + 1.53·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5736665932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5736665932\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 - 1.53T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.87T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.87T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 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
−9.697509774032243767733252134398, −9.185733023334604855733027574923, −8.610667495590115767976512760706, −7.52402586814344603989073733040, −6.97416443656283533655506106178, −6.05535709546618264787834007193, −5.19440990583355660462947601039, −3.25918029419843707722458492791, −2.21724250953410198289258372309, −1.19443623302098556629034947050,
1.19443623302098556629034947050, 2.21724250953410198289258372309, 3.25918029419843707722458492791, 5.19440990583355660462947601039, 6.05535709546618264787834007193, 6.97416443656283533655506106178, 7.52402586814344603989073733040, 8.610667495590115767976512760706, 9.185733023334604855733027574923, 9.697509774032243767733252134398
