| L(s) = 1 | + 1.36·2-s + 3-s − 0.135·4-s + 1.32·5-s + 1.36·6-s − 2.91·8-s + 9-s + 1.81·10-s − 1.42·11-s − 0.135·12-s − 3.16·13-s + 1.32·15-s − 3.71·16-s − 1.15·17-s + 1.36·18-s − 8.57·19-s − 0.179·20-s − 1.93·22-s − 23-s − 2.91·24-s − 3.24·25-s − 4.32·26-s + 27-s + 7.45·29-s + 1.81·30-s − 1.86·31-s + 0.764·32-s + ⋯ |
| L(s) = 1 | + 0.965·2-s + 0.577·3-s − 0.0677·4-s + 0.593·5-s + 0.557·6-s − 1.03·8-s + 0.333·9-s + 0.572·10-s − 0.428·11-s − 0.0391·12-s − 0.878·13-s + 0.342·15-s − 0.927·16-s − 0.281·17-s + 0.321·18-s − 1.96·19-s − 0.0401·20-s − 0.413·22-s − 0.208·23-s − 0.595·24-s − 0.648·25-s − 0.847·26-s + 0.192·27-s + 1.38·29-s + 0.330·30-s − 0.335·31-s + 0.135·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 11 | \( 1 + 1.42T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 + 8.57T + 19T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 6.35T + 89T^{2} \) |
| 97 | \( 1 + 0.408T + 97T^{2} \) |
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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
−8.446770245773057304755852691149, −7.43062541705693924565345726794, −6.48782081227814596794043409563, −5.96715368491096569837428220814, −4.93064367168029095955759109537, −4.49812098952554220284091087815, −3.56445822030712101905626738097, −2.62679951457734620048668512692, −1.96571166549381669773357646782, 0,
1.96571166549381669773357646782, 2.62679951457734620048668512692, 3.56445822030712101905626738097, 4.49812098952554220284091087815, 4.93064367168029095955759109537, 5.96715368491096569837428220814, 6.48782081227814596794043409563, 7.43062541705693924565345726794, 8.446770245773057304755852691149
