| L(s) = 1 | + 2-s − 1.68·3-s + 4-s − 2.87·5-s − 1.68·6-s − 3.82·7-s + 8-s − 0.163·9-s − 2.87·10-s − 1.07·11-s − 1.68·12-s + 3.46·13-s − 3.82·14-s + 4.83·15-s + 16-s − 0.673·17-s − 0.163·18-s + 4.31·19-s − 2.87·20-s + 6.43·21-s − 1.07·22-s + 4.95·23-s − 1.68·24-s + 3.25·25-s + 3.46·26-s + 5.32·27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.972·3-s + 0.5·4-s − 1.28·5-s − 0.687·6-s − 1.44·7-s + 0.353·8-s − 0.0545·9-s − 0.908·10-s − 0.322·11-s − 0.486·12-s + 0.961·13-s − 1.02·14-s + 1.24·15-s + 0.250·16-s − 0.163·17-s − 0.0385·18-s + 0.988·19-s − 0.642·20-s + 1.40·21-s − 0.228·22-s + 1.03·23-s − 0.343·24-s + 0.650·25-s + 0.679·26-s + 1.02·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 0.673T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 0.558T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 6.99T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.74T + 73T^{2} \) |
| 79 | \( 1 - 9.83T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.51254979094659926687828493637, −6.66840046775937855790247073950, −6.45738995195846243936331006641, −5.44934802332985423690117253554, −5.00815458368079215739358284500, −3.91198658669826587836357376398, −3.42475704087377667128117683363, −2.76208354363140564768168073625, −1.00292837683658757320764347062, 0,
1.00292837683658757320764347062, 2.76208354363140564768168073625, 3.42475704087377667128117683363, 3.91198658669826587836357376398, 5.00815458368079215739358284500, 5.44934802332985423690117253554, 6.45738995195846243936331006641, 6.66840046775937855790247073950, 7.51254979094659926687828493637
