| L(s) = 1 | − 0.826·2-s − 0.332·3-s − 1.31·4-s − 4.12·5-s + 0.274·6-s − 3.59·7-s + 2.74·8-s − 2.88·9-s + 3.41·10-s + 4.39·11-s + 0.437·12-s + 2.97·14-s + 1.37·15-s + 0.365·16-s − 6.86·17-s + 2.38·18-s − 3.97·19-s + 5.43·20-s + 1.19·21-s − 3.63·22-s − 2.31·23-s − 0.911·24-s + 12.0·25-s + 1.95·27-s + 4.73·28-s − 7.05·29-s − 1.13·30-s + ⋯ |
| L(s) = 1 | − 0.584·2-s − 0.191·3-s − 0.658·4-s − 1.84·5-s + 0.112·6-s − 1.35·7-s + 0.969·8-s − 0.963·9-s + 1.07·10-s + 1.32·11-s + 0.126·12-s + 0.794·14-s + 0.354·15-s + 0.0914·16-s − 1.66·17-s + 0.563·18-s − 0.911·19-s + 1.21·20-s + 0.260·21-s − 0.775·22-s − 0.482·23-s − 0.186·24-s + 2.40·25-s + 0.376·27-s + 0.894·28-s − 1.30·29-s − 0.207·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.826T + 2T^{2} \) |
| 3 | \( 1 + 0.332T + 3T^{2} \) |
| 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 19 | \( 1 + 3.97T + 19T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 + 7.05T + 29T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 2.95T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 - 7.56T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 9.57T + 83T^{2} \) |
| 89 | \( 1 - 0.170T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.116632708214288608645578745221, −7.08667986677559650569252428385, −6.66904399252021851524062402157, −5.81595319659124409386783708515, −4.60401985960688961143675416662, −3.94414766896913945411122083799, −3.66625125088079569740879709585, −2.43014270997479539307458879297, −0.68275234763840572630823295377, 0,
0.68275234763840572630823295377, 2.43014270997479539307458879297, 3.66625125088079569740879709585, 3.94414766896913945411122083799, 4.60401985960688961143675416662, 5.81595319659124409386783708515, 6.66904399252021851524062402157, 7.08667986677559650569252428385, 8.116632708214288608645578745221
