| L(s) = 1 | − 1.17·2-s + 3-s − 0.628·4-s − 2.94·5-s − 1.17·6-s + 3.07·8-s + 9-s + 3.45·10-s + 0.735·11-s − 0.628·12-s − 4.56·13-s − 2.94·15-s − 2.34·16-s + 1.29·17-s − 1.17·18-s + 0.205·19-s + 1.85·20-s − 0.860·22-s + 7.81·23-s + 3.07·24-s + 3.68·25-s + 5.34·26-s + 27-s − 10.2·29-s + 3.45·30-s + 6.98·31-s − 3.40·32-s + ⋯ |
| L(s) = 1 | − 0.827·2-s + 0.577·3-s − 0.314·4-s − 1.31·5-s − 0.478·6-s + 1.08·8-s + 0.333·9-s + 1.09·10-s + 0.221·11-s − 0.181·12-s − 1.26·13-s − 0.761·15-s − 0.586·16-s + 0.315·17-s − 0.275·18-s + 0.0470·19-s + 0.414·20-s − 0.183·22-s + 1.63·23-s + 0.628·24-s + 0.737·25-s + 1.04·26-s + 0.192·27-s − 1.89·29-s + 0.630·30-s + 1.25·31-s − 0.602·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 + 2.94T + 5T^{2} \) |
| 11 | \( 1 - 0.735T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 - 0.205T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 3.34T + 47T^{2} \) |
| 53 | \( 1 + 5.83T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 0.636T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 0.379T + 97T^{2} \) |
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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
−7.80700109725684555011390912135, −7.30391521511802860284431002893, −6.83060368867053177777658744436, −5.31230692506470604328725959850, −4.75228782590993748529960138389, −3.94484969965356934004451745594, −3.32298595179586843536210580489, −2.24261069131375314008535860420, −1.03806998185538114544352782578, 0,
1.03806998185538114544352782578, 2.24261069131375314008535860420, 3.32298595179586843536210580489, 3.94484969965356934004451745594, 4.75228782590993748529960138389, 5.31230692506470604328725959850, 6.83060368867053177777658744436, 7.30391521511802860284431002893, 7.80700109725684555011390912135
