| L(s) = 1 | − 2.06·2-s + 0.566·3-s + 2.27·4-s + 0.827·5-s − 1.17·6-s + 0.908·7-s − 0.574·8-s − 2.67·9-s − 1.71·10-s + 3.66·11-s + 1.29·12-s − 3.44·13-s − 1.87·14-s + 0.468·15-s − 3.36·16-s + 5.13·17-s + 5.54·18-s − 1.92·19-s + 1.88·20-s + 0.514·21-s − 7.57·22-s + 4.20·23-s − 0.325·24-s − 4.31·25-s + 7.12·26-s − 3.21·27-s + 2.06·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 0.327·3-s + 1.13·4-s + 0.369·5-s − 0.478·6-s + 0.343·7-s − 0.203·8-s − 0.893·9-s − 0.541·10-s + 1.10·11-s + 0.372·12-s − 0.955·13-s − 0.502·14-s + 0.120·15-s − 0.841·16-s + 1.24·17-s + 1.30·18-s − 0.440·19-s + 0.421·20-s + 0.112·21-s − 1.61·22-s + 0.877·23-s − 0.0664·24-s − 0.863·25-s + 1.39·26-s − 0.619·27-s + 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 - 0.566T + 3T^{2} \) |
| 5 | \( 1 - 0.827T + 5T^{2} \) |
| 7 | \( 1 - 0.908T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 1.92T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 + 3.28T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 - 4.69T + 67T^{2} \) |
| 71 | \( 1 + 4.16T + 71T^{2} \) |
| 73 | \( 1 + 7.60T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 + 8.21T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.140375613408204510493163350897, −7.59658641541318059243018917257, −6.96816986941970165598754495783, −5.98132886644251283850629727877, −5.27235449943899458978679191338, −4.16073824886017964080913206448, −3.12204786566288803316743148997, −2.10210759323851575469273784204, −1.35326694447503173323596002467, 0,
1.35326694447503173323596002467, 2.10210759323851575469273784204, 3.12204786566288803316743148997, 4.16073824886017964080913206448, 5.27235449943899458978679191338, 5.98132886644251283850629727877, 6.96816986941970165598754495783, 7.59658641541318059243018917257, 8.140375613408204510493163350897
