| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 7-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s − 14-s − 2·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s + 4·22-s − 24-s − 25-s − 2·26-s − 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.94629913909570, −14.91645422974606, −14.70511663335651, −14.26666570822075, −13.56523195305648, −13.07651711751556, −12.53726012898579, −12.04014633124826, −11.65013418166426, −10.85666833337042, −10.36510608100660, −9.731700885058634, −9.402361832095806, −8.620907730304785, −7.760368965373747, −7.089204678564828, −6.548874230996146, −6.091038872538855, −5.400769332064090, −5.110715929281469, −4.037922192580933, −3.676426367878618, −2.788842719166815, −1.826521200014953, −1.382486594438415, 0,
1.382486594438415, 1.826521200014953, 2.788842719166815, 3.676426367878618, 4.037922192580933, 5.110715929281469, 5.400769332064090, 6.091038872538855, 6.548874230996146, 7.089204678564828, 7.760368965373747, 8.620907730304785, 9.402361832095806, 9.731700885058634, 10.36510608100660, 10.85666833337042, 11.65013418166426, 12.04014633124826, 12.53726012898579, 13.07651711751556, 13.56523195305648, 14.26666570822075, 14.70511663335651, 14.91645422974606, 15.94629913909570
