L(s) = 1 | + (−2 + 3.46i)5-s + (−2.5 + 0.866i)7-s − 3·13-s + (2 + 3.46i)17-s + (3.5 − 6.06i)19-s + (2 − 3.46i)23-s + (−5.49 − 9.52i)25-s − 8·29-s + (−2.5 − 4.33i)31-s + (2.00 − 10.3i)35-s + (−1.5 + 2.59i)37-s + 8·41-s − 11·43-s + (2 − 3.46i)47-s + (5.5 − 4.33i)49-s + ⋯ |
L(s) = 1 | + (−0.894 + 1.54i)5-s + (−0.944 + 0.327i)7-s − 0.832·13-s + (0.485 + 0.840i)17-s + (0.802 − 1.39i)19-s + (0.417 − 0.722i)23-s + (−1.09 − 1.90i)25-s − 1.48·29-s + (−0.449 − 0.777i)31-s + (0.338 − 1.75i)35-s + (−0.246 + 0.427i)37-s + 1.24·41-s − 1.67·43-s + (0.291 − 0.505i)47-s + (0.785 − 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.816837575317312139031826969606, −8.916831033182143863108953580597, −7.73249703539176369559659548647, −7.13175620284314832729278167324, −6.50082714280516506194874059167, −5.45767443891443563886328149961, −4.07473204626147215665434951344, −3.19798615536227075639595940484, −2.49061059316318362188275236199, 0,
1.35666611484060859701475535105, 3.24326086864043172269554851350, 4.01214972767985722727138792610, 5.09011708905491618833734717326, 5.71629900459311173900762353035, 7.28463351284945269848124155424, 7.59601618758351430060709613098, 8.672572862635385294877753235305, 9.488019967332540966935454192217