| L(s) = 1 | + 2-s − 1.49·3-s + 4-s − 0.207·5-s − 1.49·6-s + 7-s + 8-s − 0.758·9-s − 0.207·10-s + 1.25·11-s − 1.49·12-s − 1.61·13-s + 14-s + 0.310·15-s + 16-s + 0.634·17-s − 0.758·18-s − 0.170·19-s − 0.207·20-s − 1.49·21-s + 1.25·22-s − 2.22·23-s − 1.49·24-s − 4.95·25-s − 1.61·26-s + 5.62·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.864·3-s + 0.5·4-s − 0.0928·5-s − 0.611·6-s + 0.377·7-s + 0.353·8-s − 0.252·9-s − 0.0656·10-s + 0.377·11-s − 0.432·12-s − 0.447·13-s + 0.267·14-s + 0.0802·15-s + 0.250·16-s + 0.153·17-s − 0.178·18-s − 0.0390·19-s − 0.0464·20-s − 0.326·21-s + 0.267·22-s − 0.464·23-s − 0.305·24-s − 0.991·25-s − 0.316·26-s + 1.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 0.207T + 5T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 0.634T + 17T^{2} \) |
| 19 | \( 1 + 0.170T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 8.15T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.56T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 6.59T + 79T^{2} \) |
| 83 | \( 1 - 1.90T + 83T^{2} \) |
| 89 | \( 1 + 8.58T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.58526046594372176059718837788, −6.80964505807584341235984434425, −6.14550244425935069270863624370, −5.55497475064576638125306854584, −4.90609684793842261744782535817, −4.22360123132971122697608247458, −3.35180224043502099092142013210, −2.39466334271068222351769697563, −1.36587487645259048412144703851, 0,
1.36587487645259048412144703851, 2.39466334271068222351769697563, 3.35180224043502099092142013210, 4.22360123132971122697608247458, 4.90609684793842261744782535817, 5.55497475064576638125306854584, 6.14550244425935069270863624370, 6.80964505807584341235984434425, 7.58526046594372176059718837788
