L(s) = 1 | + 4·7-s + 3·11-s + (−1 − 1.73i)13-s + (−3 + 5.19i)17-s + (3.5 + 2.59i)19-s + (3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 2·31-s − 10·37-s + (4.5 − 7.79i)41-s + (−2 + 3.46i)43-s + 9·49-s + (3 + 5.19i)53-s + (4.5 − 7.79i)59-s + (2 + 3.46i)61-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 0.904·11-s + (−0.277 − 0.480i)13-s + (−0.727 + 1.26i)17-s + (0.802 + 0.596i)19-s + (0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 0.359·31-s − 1.64·37-s + (0.702 − 1.21i)41-s + (−0.304 + 0.528i)43-s + 1.28·49-s + (0.412 + 0.713i)53-s + (0.585 − 1.01i)59-s + (0.256 + 0.443i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305641188\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305641188\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.866965424272802510991259639094, −8.130877166547134200525843133549, −7.47048054528617358571003891896, −6.73228229957901982452071099600, −5.58974977633622392681504014934, −5.14055131386955070080165044730, −4.11220154871372790066642210530, −3.38410425171896998797815321964, −1.92066422338471273666688719567, −1.28057964645246709468571462560,
0.846566516559406949925808136369, 1.96306291386956107663605922492, 2.90224816794918410657245787754, 4.25561498081208946276202095250, 4.76003577771701243787518705418, 5.44641619484999049912524165011, 6.81407231652924487367687030366, 7.01474542960158354985191035048, 8.087322983214289735111935441254, 8.809842204645031191082320694440