| L(s) = 1 | − 3.65·2-s + 3·3-s + 5.34·4-s + 11.9·5-s − 10.9·6-s + 9.68·8-s + 9·9-s − 43.5·10-s − 11·11-s + 16.0·12-s − 82.7·13-s + 35.7·15-s − 78.1·16-s + 98.8·17-s − 32.8·18-s − 97.0·19-s + 63.8·20-s + 40.1·22-s + 177.·23-s + 29.0·24-s + 17.3·25-s + 302.·26-s + 27·27-s + 123.·29-s − 130.·30-s − 208.·31-s + 208.·32-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.668·4-s + 1.06·5-s − 0.745·6-s + 0.428·8-s + 0.333·9-s − 1.37·10-s − 0.301·11-s + 0.385·12-s − 1.76·13-s + 0.616·15-s − 1.22·16-s + 1.41·17-s − 0.430·18-s − 1.17·19-s + 0.713·20-s + 0.389·22-s + 1.60·23-s + 0.247·24-s + 0.138·25-s + 2.27·26-s + 0.192·27-s + 0.792·29-s − 0.795·30-s − 1.20·31-s + 1.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.65T + 8T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 13 | \( 1 + 82.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 218.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 161.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 639.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 456.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 113.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 235.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 435.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−8.840375696861498690791192097064, −7.913750203065746370472171977609, −7.36780902537482382693492419956, −6.52564674078460459872163072538, −5.31838908033877286039468123815, −4.60069037853321784757147721769, −3.03233047289325728015266743927, −2.18632794480250948746434363449, −1.32691143235336719384527446645, 0,
1.32691143235336719384527446645, 2.18632794480250948746434363449, 3.03233047289325728015266743927, 4.60069037853321784757147721769, 5.31838908033877286039468123815, 6.52564674078460459872163072538, 7.36780902537482382693492419956, 7.913750203065746370472171977609, 8.840375696861498690791192097064
