L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 6·13-s + 16-s − 17-s + 4·19-s − 20-s − 22-s + 25-s − 6·26-s − 6·29-s − 32-s + 34-s − 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + 44-s + 8·47-s − 7·49-s − 50-s + 6·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 1.16·47-s − 49-s − 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.20916630671423, −15.59291439225709, −15.43839484156812, −14.50501619749561, −14.04455426680990, −13.39226085972240, −12.79877452297956, −12.20592562971697, −11.40459122346186, −11.22050518799189, −10.65802749273998, −9.919653178054412, −9.164493268754505, −8.956983650365216, −8.119014706168677, −7.712762118795212, −7.050896144251517, −6.299470393549461, −5.881455916175576, −5.012716301554793, −4.118597161963443, −3.505139891636901, −2.882878046777655, −1.701675196050300, −1.149203447131931, 0,
1.149203447131931, 1.701675196050300, 2.882878046777655, 3.505139891636901, 4.118597161963443, 5.012716301554793, 5.881455916175576, 6.299470393549461, 7.050896144251517, 7.712762118795212, 8.119014706168677, 8.956983650365216, 9.164493268754505, 9.919653178054412, 10.65802749273998, 11.22050518799189, 11.40459122346186, 12.20592562971697, 12.79877452297956, 13.39226085972240, 14.04455426680990, 14.50501619749561, 15.43839484156812, 15.59291439225709, 16.20916630671423