| L(s) = 1 | + 0.291·2-s + 2.43·3-s − 1.91·4-s − 2.06·5-s + 0.708·6-s − 1.14·8-s + 2.91·9-s − 0.601·10-s − 4.94·11-s − 4.65·12-s + 1.62·13-s − 5.02·15-s + 3.49·16-s − 6.99·17-s + 0.849·18-s + 19-s + 3.95·20-s − 1.44·22-s − 7.41·23-s − 2.77·24-s − 0.732·25-s + 0.473·26-s − 0.206·27-s + 3.88·29-s − 1.46·30-s + 0.808·31-s + 3.30·32-s + ⋯ |
| L(s) = 1 | + 0.206·2-s + 1.40·3-s − 0.957·4-s − 0.923·5-s + 0.289·6-s − 0.403·8-s + 0.971·9-s − 0.190·10-s − 1.49·11-s − 1.34·12-s + 0.450·13-s − 1.29·15-s + 0.874·16-s − 1.69·17-s + 0.200·18-s + 0.229·19-s + 0.884·20-s − 0.307·22-s − 1.54·23-s − 0.566·24-s − 0.146·25-s + 0.0927·26-s − 0.0397·27-s + 0.721·29-s − 0.267·30-s + 0.145·31-s + 0.583·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.291T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 0.808T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.51T + 71T^{2} \) |
| 73 | \( 1 + 2.08T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 6.69T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−9.461284297428398745015148974455, −8.593085408067978987643885427231, −8.126640982739596599643063355912, −7.60102029932519557285196284378, −6.18248209081781615462126075373, −4.87061666165091352407650187698, −4.11297018263441469232145835419, −3.30840282415389705541618557284, −2.27205231617264581194724814111, 0,
2.27205231617264581194724814111, 3.30840282415389705541618557284, 4.11297018263441469232145835419, 4.87061666165091352407650187698, 6.18248209081781615462126075373, 7.60102029932519557285196284378, 8.126640982739596599643063355912, 8.593085408067978987643885427231, 9.461284297428398745015148974455
