| L(s) = 1 | − 3.14·3-s + 2.58·5-s − 1.35·7-s + 6.86·9-s + 5.76·11-s − 3.53·13-s − 8.10·15-s + 7.42·17-s − 1.86·19-s + 4.25·21-s − 2.72·23-s + 1.65·25-s − 12.1·27-s + 4.67·29-s − 4.18·31-s − 18.0·33-s − 3.49·35-s − 3.82·37-s + 11.1·39-s − 4.08·41-s − 4.41·43-s + 17.7·45-s − 4.01·47-s − 5.16·49-s − 23.3·51-s − 12.7·53-s + 14.8·55-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 1.15·5-s − 0.511·7-s + 2.28·9-s + 1.73·11-s − 0.980·13-s − 2.09·15-s + 1.80·17-s − 0.427·19-s + 0.927·21-s − 0.569·23-s + 0.331·25-s − 2.33·27-s + 0.867·29-s − 0.751·31-s − 3.14·33-s − 0.590·35-s − 0.629·37-s + 1.77·39-s − 0.638·41-s − 0.673·43-s + 2.63·45-s − 0.586·47-s − 0.738·49-s − 3.26·51-s − 1.75·53-s + 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 9.86T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 8.26T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.03193268646271865087606033887, −6.62092742450784446380152133038, −6.08120982793157119028974850850, −5.53377565329172603590178519416, −4.94091677769835836404666960550, −4.10741236765752691074983844023, −3.20534746548187741072489843006, −1.77396614991340470231903155364, −1.25561470854891322877246887745, 0,
1.25561470854891322877246887745, 1.77396614991340470231903155364, 3.20534746548187741072489843006, 4.10741236765752691074983844023, 4.94091677769835836404666960550, 5.53377565329172603590178519416, 6.08120982793157119028974850850, 6.62092742450784446380152133038, 7.03193268646271865087606033887
