| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 7-s + 4·10-s + 11-s − 7·13-s − 2·14-s − 4·16-s − 5·17-s − 19-s − 4·20-s − 2·22-s − 4·23-s − 25-s + 14·26-s + 2·28-s − 2·29-s + 4·31-s + 8·32-s + 10·34-s − 2·35-s + 4·37-s + 2·38-s − 4·41-s + 12·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 0.377·7-s + 1.26·10-s + 0.301·11-s − 1.94·13-s − 0.534·14-s − 16-s − 1.21·17-s − 0.229·19-s − 0.894·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 2.74·26-s + 0.377·28-s − 0.371·29-s + 0.718·31-s + 1.41·32-s + 1.71·34-s − 0.338·35-s + 0.657·37-s + 0.324·38-s − 0.624·41-s + 1.82·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13167 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13167 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.81471350279746, −15.96790383862958, −15.58138442781850, −15.07364736675767, −14.36157113950744, −13.86527012019480, −13.01058600794892, −12.27959624422420, −11.86166452538655, −11.21101814209408, −10.82950665681270, −9.967658402341313, −9.644329515184468, −8.980225087818698, −8.347253909569436, −7.815307277563320, −7.379948272668275, −6.837542431368537, −6.024971389173359, −4.867410270697395, −4.484614445796865, −3.738037504819214, −2.380903368374912, −2.122835830751390, −0.7749709379148627, 0,
0.7749709379148627, 2.122835830751390, 2.380903368374912, 3.738037504819214, 4.484614445796865, 4.867410270697395, 6.024971389173359, 6.837542431368537, 7.379948272668275, 7.815307277563320, 8.347253909569436, 8.980225087818698, 9.644329515184468, 9.967658402341313, 10.82950665681270, 11.21101814209408, 11.86166452538655, 12.27959624422420, 13.01058600794892, 13.86527012019480, 14.36157113950744, 15.07364736675767, 15.58138442781850, 15.96790383862958, 16.81471350279746
