| L(s) = 1 | − 2-s − 2.79·3-s + 4-s + 5-s + 2.79·6-s + 1.79·7-s − 8-s + 4.79·9-s − 10-s + 0.791·11-s − 2.79·12-s + 5.79·13-s − 1.79·14-s − 2.79·15-s + 16-s − 0.791·17-s − 4.79·18-s − 5.79·19-s + 20-s − 5·21-s − 0.791·22-s + 2.79·24-s + 25-s − 5.79·26-s − 4.99·27-s + 1.79·28-s + 7.58·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.447·5-s + 1.13·6-s + 0.677·7-s − 0.353·8-s + 1.59·9-s − 0.316·10-s + 0.238·11-s − 0.805·12-s + 1.60·13-s − 0.478·14-s − 0.720·15-s + 0.250·16-s − 0.191·17-s − 1.12·18-s − 1.32·19-s + 0.223·20-s − 1.09·21-s − 0.168·22-s + 0.569·24-s + 0.200·25-s − 1.13·26-s − 0.962·27-s + 0.338·28-s + 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 0.791T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
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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
−7.983632026711558776898517608607, −6.77682055724084887237329671925, −6.39207443516318198282613810262, −5.95634571640188433173166948228, −4.98346214175879970956748007345, −4.43479227425195452680727738060, −3.26541588307983695276272334365, −1.78526885550021860631648280189, −1.25733118263330262111971044933, 0,
1.25733118263330262111971044933, 1.78526885550021860631648280189, 3.26541588307983695276272334365, 4.43479227425195452680727738060, 4.98346214175879970956748007345, 5.95634571640188433173166948228, 6.39207443516318198282613810262, 6.77682055724084887237329671925, 7.983632026711558776898517608607
