| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 17-s + 2·18-s − 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 5·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.839452344153785115520873379088, −8.056387257148129083410786981602, −7.12157505509133992034130528247, −6.27294138686605894810767727005, −5.83790014329485352231929095187, −4.86594603190401980356996477627, −3.58326559945619076008065110726, −2.60397272983966880463460367895, −1.38736378691258344967843613523, 0,
1.38736378691258344967843613523, 2.60397272983966880463460367895, 3.58326559945619076008065110726, 4.86594603190401980356996477627, 5.83790014329485352231929095187, 6.27294138686605894810767727005, 7.12157505509133992034130528247, 8.056387257148129083410786981602, 8.839452344153785115520873379088
