L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (−2 − 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (1 − 1.73i)11-s + (0.499 + 0.866i)12-s + 13-s + (0.499 − 2.59i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.301 − 0.522i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (0.133 − 0.694i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.121 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10045 - 0.837179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10045 - 0.837179i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.21110771005934492187322117628, −9.050796845443754152158729833856, −8.450804944623367116934427594453, −7.58554537764802572585635106350, −6.54134813905353322593771430461, −6.17020652367192246136571630122, −4.64721617978732320266774091698, −3.84561735311673590311663966993, −2.63698772945831770356222991004, −0.61377756505634290878075048409,
1.91815659072339918073338665907, 3.17758821435670594168676653381, 3.82326114511984173412715042913, 5.04263165218809237201221764705, 6.01606544434852511983950118310, 6.98517740600275583857247075300, 8.223840495622797846490677203246, 9.138443287715048141997337253603, 9.857310598643756610101947839672, 10.47946905391854915008941345020