| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s + 1.24·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s + 1.55·10-s + 2·11-s − 0.999·12-s − 2.24·15-s − 1.24·16-s + 2.80·18-s − 1.80·19-s + 0.692·20-s + 2.49·22-s − 0.445·23-s + 1.00·24-s + 0.554·25-s − 2.24·27-s − 0.445·29-s − 2.80·30-s − 0.999·32-s − 3.60·33-s + 1.24·36-s − 2.24·38-s − 0.692·40-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s + 1.24·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s + 1.55·10-s + 2·11-s − 0.999·12-s − 2.24·15-s − 1.24·16-s + 2.80·18-s − 1.80·19-s + 0.692·20-s + 2.49·22-s − 0.445·23-s + 1.00·24-s + 0.554·25-s − 2.24·27-s − 0.445·29-s − 2.80·30-s − 0.999·32-s − 3.60·33-s + 1.24·36-s − 2.24·38-s − 0.692·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.072132799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072132799\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 2T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 - 1.24T + T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−11.61321396543275839166813004970, −10.80695901122252835963209850005, −9.806850504849358835341516641413, −8.965703322197701109812993718759, −6.79738973199921560514862087173, −6.24798536474062340609840084125, −5.80794320174755592041415851401, −4.71072816950501530034895440899, −3.95465751271839231881092563948, −1.78247962675410925792230700719,
1.78247962675410925792230700719, 3.95465751271839231881092563948, 4.71072816950501530034895440899, 5.80794320174755592041415851401, 6.24798536474062340609840084125, 6.79738973199921560514862087173, 8.965703322197701109812993718759, 9.806850504849358835341516641413, 10.80695901122252835963209850005, 11.61321396543275839166813004970
