| L(s) = 1 | + 1.55·2-s + 1.14·3-s + 0.403·4-s + 1.76·6-s − 3.74·7-s − 2.47·8-s − 1.69·9-s + 1.45·11-s + 0.460·12-s + 6.71·13-s − 5.80·14-s − 4.64·16-s − 2.63·18-s + 1.87·19-s − 4.26·21-s + 2.26·22-s + 4.23·23-s − 2.82·24-s + 10.4·26-s − 5.35·27-s − 1.51·28-s − 7.59·29-s + 1.29·31-s − 2.25·32-s + 1.66·33-s − 0.686·36-s + 4.51·37-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.658·3-s + 0.201·4-s + 0.721·6-s − 1.41·7-s − 0.874·8-s − 0.566·9-s + 0.439·11-s + 0.132·12-s + 1.86·13-s − 1.55·14-s − 1.16·16-s − 0.621·18-s + 0.430·19-s − 0.931·21-s + 0.482·22-s + 0.882·23-s − 0.576·24-s + 2.04·26-s − 1.03·27-s − 0.285·28-s − 1.41·29-s + 0.232·31-s − 0.397·32-s + 0.289·33-s − 0.114·36-s + 0.742·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.216140717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.216140717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 - 6.71T + 13T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.036945706618238376839166129996, −6.87073106851324362494671545729, −6.40923642280709628976475290500, −5.78173750268969340710209215779, −5.18898243992076567155044782750, −3.97268609126863530988445930188, −3.52207068424968039267508214681, −3.19674127405414687490376213808, −2.20548120233696715422861127958, −0.72617151134386444405438193907,
0.72617151134386444405438193907, 2.20548120233696715422861127958, 3.19674127405414687490376213808, 3.52207068424968039267508214681, 3.97268609126863530988445930188, 5.18898243992076567155044782750, 5.78173750268969340710209215779, 6.40923642280709628976475290500, 6.87073106851324362494671545729, 8.036945706618238376839166129996
