| L(s) = 1 | − 0.700·2-s − 2.17·3-s − 1.50·4-s − 0.553·5-s + 1.52·6-s − 0.939·7-s + 2.45·8-s + 1.72·9-s + 0.387·10-s + 4.19·11-s + 3.28·12-s + 2.15·13-s + 0.657·14-s + 1.20·15-s + 1.29·16-s + 17-s − 1.20·18-s + 1.29·19-s + 0.835·20-s + 2.04·21-s − 2.93·22-s − 5.86·23-s − 5.34·24-s − 4.69·25-s − 1.51·26-s + 2.76·27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.495·2-s − 1.25·3-s − 0.754·4-s − 0.247·5-s + 0.621·6-s − 0.355·7-s + 0.868·8-s + 0.575·9-s + 0.122·10-s + 1.26·11-s + 0.947·12-s + 0.598·13-s + 0.175·14-s + 0.310·15-s + 0.324·16-s + 0.242·17-s − 0.284·18-s + 0.296·19-s + 0.186·20-s + 0.445·21-s − 0.625·22-s − 1.22·23-s − 1.09·24-s − 0.938·25-s − 0.296·26-s + 0.532·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.700T + 2T^{2} \) |
| 3 | \( 1 + 2.17T + 3T^{2} \) |
| 5 | \( 1 + 0.553T + 5T^{2} \) |
| 7 | \( 1 + 0.939T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 0.839T + 53T^{2} \) |
| 59 | \( 1 + 2.69T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 0.826T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−9.868215953477435377684843491803, −9.304420421978088070875808989660, −8.303041099844905163021530612832, −7.37140447969331917375657470324, −6.22288960371893637357964683010, −5.66438956674872378889686427438, −4.42579941029404666188292525367, −3.68098274390242604583991817724, −1.37808404077125745444183310172, 0,
1.37808404077125745444183310172, 3.68098274390242604583991817724, 4.42579941029404666188292525367, 5.66438956674872378889686427438, 6.22288960371893637357964683010, 7.37140447969331917375657470324, 8.303041099844905163021530612832, 9.304420421978088070875808989660, 9.868215953477435377684843491803
