| L(s) = 1 | − 2.03·2-s + 1.78·3-s + 2.15·4-s − 0.701·5-s − 3.64·6-s − 2.02·7-s − 0.325·8-s + 0.187·9-s + 1.42·10-s − 0.626·11-s + 3.85·12-s − 2.93·13-s + 4.13·14-s − 1.25·15-s − 3.65·16-s − 2.71·17-s − 0.382·18-s − 1.11·19-s − 1.51·20-s − 3.62·21-s + 1.27·22-s − 0.412·23-s − 0.580·24-s − 4.50·25-s + 5.99·26-s − 5.02·27-s − 4.38·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.03·3-s + 1.07·4-s − 0.313·5-s − 1.48·6-s − 0.767·7-s − 0.114·8-s + 0.0624·9-s + 0.452·10-s − 0.189·11-s + 1.11·12-s − 0.814·13-s + 1.10·14-s − 0.323·15-s − 0.913·16-s − 0.657·17-s − 0.0900·18-s − 0.254·19-s − 0.338·20-s − 0.790·21-s + 0.272·22-s − 0.0859·23-s − 0.118·24-s − 0.901·25-s + 1.17·26-s − 0.966·27-s − 0.828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 3 | \( 1 - 1.78T + 3T^{2} \) |
| 5 | \( 1 + 0.701T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 0.412T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 + 0.135T + 41T^{2} \) |
| 43 | \( 1 + 0.861T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 + 0.406T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 2.78T + 71T^{2} \) |
| 73 | \( 1 + 2.11T + 73T^{2} \) |
| 79 | \( 1 + 0.317T + 79T^{2} \) |
| 83 | \( 1 + 5.05T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−10.09358493493378490230383544202, −9.430095301408504994635374848383, −8.790510999214242274470793903259, −7.964357785666527326501415024540, −7.33198303963524491786428404816, −6.24035767159084886716798266145, −4.51376071483096784551376712580, −3.12983915588039610610705239119, −2.07683284652257114995983312896, 0,
2.07683284652257114995983312896, 3.12983915588039610610705239119, 4.51376071483096784551376712580, 6.24035767159084886716798266145, 7.33198303963524491786428404816, 7.964357785666527326501415024540, 8.790510999214242274470793903259, 9.430095301408504994635374848383, 10.09358493493378490230383544202
