| L(s) = 1 | − 2-s + 4-s − 3.56·5-s − 7-s − 8-s + 3.56·10-s + 1.56·11-s + 13-s + 14-s + 16-s + 6.68·17-s − 4.68·19-s − 3.56·20-s − 1.56·22-s + 5.56·23-s + 7.68·25-s − 26-s − 28-s − 6.68·29-s + 6.24·31-s − 32-s − 6.68·34-s + 3.56·35-s − 7.56·37-s + 4.68·38-s + 3.56·40-s + 1.12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.59·5-s − 0.377·7-s − 0.353·8-s + 1.12·10-s + 0.470·11-s + 0.277·13-s + 0.267·14-s + 0.250·16-s + 1.62·17-s − 1.07·19-s − 0.796·20-s − 0.332·22-s + 1.15·23-s + 1.53·25-s − 0.196·26-s − 0.188·28-s − 1.24·29-s + 1.12·31-s − 0.176·32-s − 1.14·34-s + 0.602·35-s − 1.24·37-s + 0.759·38-s + 0.563·40-s + 0.175·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 7.12T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.786681186988976237867539731911, −8.269562579899544367306491342018, −7.46708607069936873467966307679, −6.89280920834416277493602414599, −5.90726272870591041979045530985, −4.69516632185330918124084363894, −3.67279003312917711289435593032, −3.06451183452278075127846203459, −1.33777036551990798132863711896, 0,
1.33777036551990798132863711896, 3.06451183452278075127846203459, 3.67279003312917711289435593032, 4.69516632185330918124084363894, 5.90726272870591041979045530985, 6.89280920834416277493602414599, 7.46708607069936873467966307679, 8.269562579899544367306491342018, 8.786681186988976237867539731911
