| L(s) = 1 | − 0.930·2-s − 1.12·3-s − 1.13·4-s + 3.65·5-s + 1.05·6-s − 1.39·7-s + 2.91·8-s − 1.72·9-s − 3.40·10-s + 1.37·11-s + 1.27·12-s − 4.81·13-s + 1.29·14-s − 4.12·15-s − 0.446·16-s − 2.61·17-s + 1.60·18-s + 3.96·19-s − 4.14·20-s + 1.57·21-s − 1.28·22-s − 5.45·23-s − 3.29·24-s + 8.36·25-s + 4.48·26-s + 5.33·27-s + 1.58·28-s + ⋯ |
| L(s) = 1 | − 0.658·2-s − 0.651·3-s − 0.566·4-s + 1.63·5-s + 0.428·6-s − 0.527·7-s + 1.03·8-s − 0.575·9-s − 1.07·10-s + 0.415·11-s + 0.369·12-s − 1.33·13-s + 0.346·14-s − 1.06·15-s − 0.111·16-s − 0.635·17-s + 0.378·18-s + 0.909·19-s − 0.926·20-s + 0.343·21-s − 0.273·22-s − 1.13·23-s − 0.671·24-s + 1.67·25-s + 0.879·26-s + 1.02·27-s + 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 + 0.930T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 + 0.992T + 29T^{2} \) |
| 31 | \( 1 - 1.16T + 31T^{2} \) |
| 37 | \( 1 + 0.503T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 7.91T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 7.26T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 - 0.319T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−9.569535682821969279214726691718, −9.150424665708998282257753995336, −8.054161932136710873818241938068, −6.93187015518367451548883890480, −6.07708820474381304907643272113, −5.36912939213684750867251699634, −4.53756956916276242185705026182, −2.86367399454981220055110092698, −1.62303028437897114600149837511, 0,
1.62303028437897114600149837511, 2.86367399454981220055110092698, 4.53756956916276242185705026182, 5.36912939213684750867251699634, 6.07708820474381304907643272113, 6.93187015518367451548883890480, 8.054161932136710873818241938068, 9.150424665708998282257753995336, 9.569535682821969279214726691718
