| L(s) = 1 | − 2-s + 3-s + 4-s − 2.22·5-s − 6-s + 1.37·7-s − 8-s + 9-s + 2.22·10-s + 3.97·11-s + 12-s − 13-s − 1.37·14-s − 2.22·15-s + 16-s + 2.05·17-s − 18-s − 6.55·19-s − 2.22·20-s + 1.37·21-s − 3.97·22-s − 0.577·23-s − 24-s − 0.0525·25-s + 26-s + 27-s + 1.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.994·5-s − 0.408·6-s + 0.518·7-s − 0.353·8-s + 0.333·9-s + 0.703·10-s + 1.19·11-s + 0.288·12-s − 0.277·13-s − 0.366·14-s − 0.574·15-s + 0.250·16-s + 0.499·17-s − 0.235·18-s − 1.50·19-s − 0.497·20-s + 0.299·21-s − 0.846·22-s − 0.120·23-s − 0.204·24-s − 0.0105·25-s + 0.196·26-s + 0.192·27-s + 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 6.55T + 19T^{2} \) |
| 23 | \( 1 + 0.577T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 6.35T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 - 9.71T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.65192221473061857629370668741, −6.99939643516900016914624587087, −6.40085959114725607214361210566, −5.43759840836449414696592236884, −4.28770160025885011767276732010, −3.99770824218729080193467872001, −3.06495735488709035301651697358, −2.06902688469757288171233503230, −1.27756694770733209873428175938, 0,
1.27756694770733209873428175938, 2.06902688469757288171233503230, 3.06495735488709035301651697358, 3.99770824218729080193467872001, 4.28770160025885011767276732010, 5.43759840836449414696592236884, 6.40085959114725607214361210566, 6.99939643516900016914624587087, 7.65192221473061857629370668741
