| L(s) = 1 | + (−2.47 − 3.14i)2-s + (−3.78 + 15.5i)4-s + (−27.2 + 27.2i)5-s + 50.3·7-s + (58.2 − 26.5i)8-s + (152. + 18.3i)10-s + (53.1 + 53.1i)11-s + (−125. − 125. i)13-s + (−124. − 158. i)14-s + (−227. − 117. i)16-s − 286.·17-s + (−99.5 + 99.5i)19-s + (−320. − 526. i)20-s + (35.7 − 298. i)22-s + 100.·23-s + ⋯ |
| L(s) = 1 | + (−0.617 − 0.786i)2-s + (−0.236 + 0.971i)4-s + (−1.08 + 1.08i)5-s + 1.02·7-s + (0.910 − 0.414i)8-s + (1.52 + 0.183i)10-s + (0.438 + 0.438i)11-s + (−0.740 − 0.740i)13-s + (−0.634 − 0.807i)14-s + (−0.888 − 0.459i)16-s − 0.990·17-s + (−0.275 + 0.275i)19-s + (−0.800 − 1.31i)20-s + (0.0739 − 0.616i)22-s + 0.189·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0846i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00161568 + 0.0380900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00161568 + 0.0380900i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.47 + 3.14i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (27.2 - 27.2i)T - 625iT^{2} \) |
| 7 | \( 1 - 50.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-53.1 - 53.1i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (125. + 125. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 286.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (99.5 - 99.5i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 - 100.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (343. + 343. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + 208. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (1.15e3 - 1.15e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 2.33e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (2.07e3 + 2.07e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 1.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (2.13e3 - 2.13e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (3.72e3 + 3.72e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (-2.49e3 - 2.49e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (329. - 329. i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.04e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.67e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.47e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (1.45e3 - 1.45e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.14e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−11.64785531783835229460806381668, −10.94726111402752637255940619795, −10.12299193946356435786054939143, −8.667487173053428896150737438407, −7.72825378049274575788849146324, −6.95861775709288589110554095352, −4.66849897485891142678809055701, −3.49247629273509656879137744282, −2.07627814804223706611539547103, −0.01924817015758369746532817381,
1.49747303424628091448160508549, 4.36182232833616897240904734185, 5.03691889071969749682294900886, 6.72368161985344067980993773944, 7.86168402160222181679249708285, 8.595079578317835189333960334058, 9.377538892490280950778628047350, 11.04450724100249654836721152220, 11.66106550700801054668938450563, 12.94655804088930733644245581624
