| L(s) = 1 | − 1.31·2-s − 0.276·4-s + 3.61·5-s − 0.837·7-s + 2.98·8-s − 4.74·10-s − 11-s + 1.11·13-s + 1.09·14-s − 3.37·16-s − 4.37·17-s + 3.54·19-s − 0.998·20-s + 1.31·22-s − 4.06·23-s + 8.06·25-s − 1.45·26-s + 0.231·28-s + 1.67·29-s − 2.81·31-s − 1.55·32-s + 5.75·34-s − 3.02·35-s + 1.86·37-s − 4.65·38-s + 10.8·40-s − 8.79·41-s + ⋯ |
| L(s) = 1 | − 0.928·2-s − 0.138·4-s + 1.61·5-s − 0.316·7-s + 1.05·8-s − 1.50·10-s − 0.301·11-s + 0.308·13-s + 0.293·14-s − 0.842·16-s − 1.06·17-s + 0.812·19-s − 0.223·20-s + 0.279·22-s − 0.847·23-s + 1.61·25-s − 0.286·26-s + 0.0437·28-s + 0.310·29-s − 0.504·31-s − 0.274·32-s + 0.986·34-s − 0.511·35-s + 0.306·37-s − 0.754·38-s + 1.70·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 0.837T + 7T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 + 8.79T + 41T^{2} \) |
| 43 | \( 1 + 0.247T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 - 0.446T + 59T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 + 0.319T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−7.997253811293623717531137346487, −6.91586901312758616481829600032, −6.49410973147690138253647808225, −5.56506436776541821018689012638, −5.04016063109797695671411320963, −4.09504302107336865169905889466, −2.96036727023722601484589915746, −1.99997276896997845658011015985, −1.36454544353000383409943910749, 0,
1.36454544353000383409943910749, 1.99997276896997845658011015985, 2.96036727023722601484589915746, 4.09504302107336865169905889466, 5.04016063109797695671411320963, 5.56506436776541821018689012638, 6.49410973147690138253647808225, 6.91586901312758616481829600032, 7.997253811293623717531137346487
