| L(s) = 1 | − 3.38·2-s + 5.26·3-s + 3.48·4-s − 18.6·5-s − 17.8·6-s + 15.2·8-s + 0.749·9-s + 63.0·10-s + 12.1·11-s + 18.3·12-s + 38.3·13-s − 98.0·15-s − 79.7·16-s − 43.6·17-s − 2.54·18-s + 151.·19-s − 64.9·20-s − 41.1·22-s − 174.·23-s + 80.5·24-s + 221.·25-s − 130.·26-s − 138.·27-s − 135.·29-s + 332.·30-s − 31·31-s + 147.·32-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 1.01·3-s + 0.436·4-s − 1.66·5-s − 1.21·6-s + 0.675·8-s + 0.0277·9-s + 1.99·10-s + 0.332·11-s + 0.442·12-s + 0.819·13-s − 1.68·15-s − 1.24·16-s − 0.622·17-s − 0.0332·18-s + 1.82·19-s − 0.725·20-s − 0.398·22-s − 1.58·23-s + 0.684·24-s + 1.76·25-s − 0.981·26-s − 0.985·27-s − 0.867·29-s + 2.02·30-s − 0.179·31-s + 0.817·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 + 31T \) |
| good | 2 | \( 1 + 3.38T + 8T^{2} \) |
| 3 | \( 1 - 5.26T + 27T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 11 | \( 1 - 12.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 225.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 63.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 183.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 646.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 674.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 213.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 557.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 803.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 712.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 179.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 276.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 942.T + 9.12e5T^{2} \) |
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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
−8.539860008594880204343001465510, −8.111303760331849439764318877134, −7.56853160900617890340757927308, −6.81585114247620988256512645508, −5.33581511990751684872406435075, −3.87774301082182195739259420850, −3.75940878699230079570244582875, −2.35271731784175814903381484513, −1.05219446888726586278635231432, 0,
1.05219446888726586278635231432, 2.35271731784175814903381484513, 3.75940878699230079570244582875, 3.87774301082182195739259420850, 5.33581511990751684872406435075, 6.81585114247620988256512645508, 7.56853160900617890340757927308, 8.111303760331849439764318877134, 8.539860008594880204343001465510
