| L(s) = 1 | + 3·11-s + 4·13-s − 3·17-s − 4·19-s − 2·23-s − 2·29-s + 4·31-s − 4·37-s − 10·41-s + 7·43-s + 4·47-s − 7·49-s + 8·53-s − 59-s + 4·61-s − 12·67-s + 14·71-s + 2·73-s + 83-s + 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 0.417·23-s − 0.371·29-s + 0.718·31-s − 0.657·37-s − 1.56·41-s + 1.06·43-s + 0.583·47-s − 49-s + 1.09·53-s − 0.130·59-s + 0.512·61-s − 1.46·67-s + 1.66·71-s + 0.234·73-s + 0.109·83-s + 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.38968875559125, −13.95302066876820, −13.47889799378432, −13.07820115148017, −12.42080235209582, −11.92914161338975, −11.49856687387492, −10.84693498177813, −10.59276850935540, −9.885907035889173, −9.291513665446706, −8.807099177673785, −8.384190202210416, −7.922557182717953, −6.959044910541980, −6.724682467292659, −6.138726561217307, −5.638134053117438, −4.869667803334284, −4.174554189536374, −3.856627361052723, −3.170679105047680, −2.303998793141186, −1.709933638037258, −0.9893344904698217, 0,
0.9893344904698217, 1.709933638037258, 2.303998793141186, 3.170679105047680, 3.856627361052723, 4.174554189536374, 4.869667803334284, 5.638134053117438, 6.138726561217307, 6.724682467292659, 6.959044910541980, 7.922557182717953, 8.384190202210416, 8.807099177673785, 9.291513665446706, 9.885907035889173, 10.59276850935540, 10.84693498177813, 11.49856687387492, 11.92914161338975, 12.42080235209582, 13.07820115148017, 13.47889799378432, 13.95302066876820, 14.38968875559125
