| L(s) = 1 | − 0.737·2-s + 0.825·3-s − 1.45·4-s − 5-s − 0.608·6-s − 7-s + 2.54·8-s − 2.31·9-s + 0.737·10-s + 0.0628·11-s − 1.20·12-s − 0.351·13-s + 0.737·14-s − 0.825·15-s + 1.03·16-s − 2.54·17-s + 1.70·18-s − 5.35·19-s + 1.45·20-s − 0.825·21-s − 0.0463·22-s + 1.58·23-s + 2.10·24-s + 25-s + 0.259·26-s − 4.38·27-s + 1.45·28-s + ⋯ |
| L(s) = 1 | − 0.521·2-s + 0.476·3-s − 0.728·4-s − 0.447·5-s − 0.248·6-s − 0.377·7-s + 0.900·8-s − 0.772·9-s + 0.233·10-s + 0.0189·11-s − 0.347·12-s − 0.0975·13-s + 0.196·14-s − 0.213·15-s + 0.258·16-s − 0.617·17-s + 0.402·18-s − 1.22·19-s + 0.325·20-s − 0.180·21-s − 0.00988·22-s + 0.331·23-s + 0.429·24-s + 0.200·25-s + 0.0508·26-s − 0.844·27-s + 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4469648502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4469648502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.737T + 2T^{2} \) |
| 3 | \( 1 - 0.825T + 3T^{2} \) |
| 11 | \( 1 - 0.0628T + 11T^{2} \) |
| 13 | \( 1 + 0.351T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 7.26T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 0.110T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−7.938794135460791939937688393428, −7.40729429896304571649509737671, −6.56736929661847399436078060625, −5.74339864650280275519285476294, −4.96202733410273774765634908644, −4.15054361556604453482994338616, −3.59150663597312221681822438286, −2.66742878370516829716564564220, −1.73171152958798928544843853340, −0.34226052472578865178839104876,
0.34226052472578865178839104876, 1.73171152958798928544843853340, 2.66742878370516829716564564220, 3.59150663597312221681822438286, 4.15054361556604453482994338616, 4.96202733410273774765634908644, 5.74339864650280275519285476294, 6.56736929661847399436078060625, 7.40729429896304571649509737671, 7.938794135460791939937688393428
