| L(s) = 1 | + 2-s − 2.44·3-s + 4-s − 5-s − 2.44·6-s − 0.745·7-s + 8-s + 2.98·9-s − 10-s − 11-s − 2.44·12-s + 2.94·13-s − 0.745·14-s + 2.44·15-s + 16-s + 3.74·17-s + 2.98·18-s − 0.153·19-s − 20-s + 1.82·21-s − 22-s − 9.39·23-s − 2.44·24-s + 25-s + 2.94·26-s + 0.0455·27-s − 0.745·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.447·5-s − 0.998·6-s − 0.281·7-s + 0.353·8-s + 0.993·9-s − 0.316·10-s − 0.301·11-s − 0.706·12-s + 0.816·13-s − 0.199·14-s + 0.631·15-s + 0.250·16-s + 0.909·17-s + 0.702·18-s − 0.0352·19-s − 0.223·20-s + 0.398·21-s − 0.213·22-s − 1.95·23-s − 0.499·24-s + 0.200·25-s + 0.577·26-s + 0.00876·27-s − 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 + 0.745T + 7T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 0.153T + 19T^{2} \) |
| 23 | \( 1 + 9.39T + 23T^{2} \) |
| 29 | \( 1 + 7.84T + 29T^{2} \) |
| 31 | \( 1 - 6.52T + 31T^{2} \) |
| 37 | \( 1 + 6.40T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 - 0.281T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 + 0.825T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 79 | \( 1 - 1.19T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.02T + 89T^{2} \) |
| 97 | \( 1 + 8.26T + 97T^{2} \) |
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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
−7.31539495743011406008525284491, −6.50600650564721520478579504956, −5.95077851296647291074215976752, −5.53832742822897184748344364610, −4.78217825593789376415849338978, −3.94772768036994730845409631222, −3.45091419168865822584452467149, −2.24018818490236484954110452131, −1.11113443069179143377843579618, 0,
1.11113443069179143377843579618, 2.24018818490236484954110452131, 3.45091419168865822584452467149, 3.94772768036994730845409631222, 4.78217825593789376415849338978, 5.53832742822897184748344364610, 5.95077851296647291074215976752, 6.50600650564721520478579504956, 7.31539495743011406008525284491
