| L(s) = 1 | + 1.28·3-s − 1.09·5-s + 0.799·7-s − 1.33·9-s − 1.15·11-s − 5.07·13-s − 1.40·15-s + 1.71·17-s + 6.34·19-s + 1.02·21-s + 8.01·23-s − 3.80·25-s − 5.59·27-s − 6.35·29-s + 5.58·31-s − 1.48·33-s − 0.873·35-s − 4.11·37-s − 6.54·39-s − 3.83·41-s + 4.86·43-s + 1.46·45-s − 7.61·47-s − 6.36·49-s + 2.21·51-s + 6.44·53-s + 1.25·55-s + ⋯ |
| L(s) = 1 | + 0.743·3-s − 0.488·5-s + 0.302·7-s − 0.446·9-s − 0.347·11-s − 1.40·13-s − 0.363·15-s + 0.416·17-s + 1.45·19-s + 0.224·21-s + 1.67·23-s − 0.761·25-s − 1.07·27-s − 1.18·29-s + 1.00·31-s − 0.258·33-s − 0.147·35-s − 0.676·37-s − 1.04·39-s − 0.598·41-s + 0.742·43-s + 0.218·45-s − 1.11·47-s − 0.908·49-s + 0.309·51-s + 0.884·53-s + 0.169·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 - 0.799T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 8.01T + 23T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 - 0.783T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.988719632187496579423971422922, −7.49915137518206808325625480262, −6.94717607601341400834372645655, −5.58144666064407208175370211117, −5.16445274674972559314741570924, −4.20379746723947277707119489663, −3.13341131027783894245139624063, −2.77279704715898858509446136609, −1.52654048744067440731071338924, 0,
1.52654048744067440731071338924, 2.77279704715898858509446136609, 3.13341131027783894245139624063, 4.20379746723947277707119489663, 5.16445274674972559314741570924, 5.58144666064407208175370211117, 6.94717607601341400834372645655, 7.49915137518206808325625480262, 7.988719632187496579423971422922
