| L(s) = 1 | − 0.231·2-s − 0.400·3-s − 1.94·4-s + 5-s + 0.0926·6-s + 1.70·7-s + 0.912·8-s − 2.83·9-s − 0.231·10-s + 0.364·11-s + 0.779·12-s − 0.500·13-s − 0.395·14-s − 0.400·15-s + 3.68·16-s − 7.94·17-s + 0.656·18-s + 0.779·19-s − 1.94·20-s − 0.684·21-s − 0.0843·22-s + 3.38·23-s − 0.365·24-s + 25-s + 0.115·26-s + 2.33·27-s − 3.32·28-s + ⋯ |
| L(s) = 1 | − 0.163·2-s − 0.231·3-s − 0.973·4-s + 0.447·5-s + 0.0378·6-s + 0.646·7-s + 0.322·8-s − 0.946·9-s − 0.0731·10-s + 0.109·11-s + 0.225·12-s − 0.138·13-s − 0.105·14-s − 0.103·15-s + 0.920·16-s − 1.92·17-s + 0.154·18-s + 0.178·19-s − 0.435·20-s − 0.149·21-s − 0.0179·22-s + 0.705·23-s − 0.0746·24-s + 0.200·25-s + 0.0226·26-s + 0.450·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 0.231T + 2T^{2} \) |
| 3 | \( 1 + 0.400T + 3T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 0.364T + 11T^{2} \) |
| 13 | \( 1 + 0.500T + 13T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 19 | \( 1 - 0.779T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 + 8.27T + 47T^{2} \) |
| 53 | \( 1 + 0.594T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 + 9.14T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.400257564612923688745760852362, −8.774685501949601766784603181296, −8.219030095208097075399036243508, −7.02717887595461921290434722962, −6.02821272710307928793436760780, −5.07550291748912268355338467949, −4.51273991461687147205305108993, −3.14614402679874471739677874155, −1.73460512528852913669178218645, 0,
1.73460512528852913669178218645, 3.14614402679874471739677874155, 4.51273991461687147205305108993, 5.07550291748912268355338467949, 6.02821272710307928793436760780, 7.02717887595461921290434722962, 8.219030095208097075399036243508, 8.774685501949601766784603181296, 9.400257564612923688745760852362
