L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−0.401 + 0.696i)5-s + (1.19 + 2.07i)7-s + 7.99·8-s + 1.60·10-s + (−29.7 − 51.5i)11-s + (−38.8 + 67.3i)13-s + (2.39 − 4.14i)14-s + (−8 − 13.8i)16-s + 2.84·17-s − 118.·19-s + (−1.60 − 2.78i)20-s + (−59.5 + 103. i)22-s + (−55.3 + 95.9i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0359 + 0.0622i)5-s + (0.0645 + 0.111i)7-s + 0.353·8-s + 0.0508·10-s + (−0.816 − 1.41i)11-s + (−0.829 + 1.43i)13-s + (0.0456 − 0.0791i)14-s + (−0.125 − 0.216i)16-s + 0.0405·17-s − 1.42·19-s + (−0.0179 − 0.0311i)20-s + (−0.577 + 0.999i)22-s + (−0.502 + 0.869i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0695688 + 0.149190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0695688 + 0.149190i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.401 - 0.696i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-1.19 - 2.07i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (29.7 + 51.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (38.8 - 67.3i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 2.84T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (55.3 - 95.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (62.8 + 108. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (31.5 - 54.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (162. - 281. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-136. - 236. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-2.46 - 4.27i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 598.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-335. + 580. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (232. + 402. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-384. + 666. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 923.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (19.9 + 34.4i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (221. + 384. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 78.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-45.5 - 78.8i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.66813979002930719105423334526, −11.52763427361620019942237009517, −10.92762246132675294537481651963, −9.742782688319014243742519388966, −8.779804123016023083445843169348, −7.80033146567465671010915497835, −6.42542566628063515798762971333, −4.93957827927447405269927582112, −3.47134011785564849430654757658, −1.99866107777991359429592635759,
0.080831565512253468667611489821, 2.34555889312077755416378264004, 4.45319584910810681867440777759, 5.49304014983334770410458792059, 6.92455721112935796467988800521, 7.78986703347105247686208000061, 8.774865336778654050781247808542, 10.24150372708628550761755222047, 10.48000588771079841833557054730, 12.41925033272218160057887343084