| L(s) = 1 | + (2.92 − 3.48i)3-s + (20.0 − 7.31i)5-s + (−36.0 − 62.3i)7-s + (10.4 + 59.4i)9-s + (55.1 − 95.5i)11-s + (−18.3 − 21.9i)13-s + (33.2 − 91.3i)15-s + (66.2 − 375. i)17-s + (−20.3 − 360. i)19-s + (−322. − 56.8i)21-s + (225. + 82.0i)23-s + (−128. + 107. i)25-s + (556. + 321. i)27-s + (1.17e3 − 206. i)29-s + (−806. + 465. i)31-s + ⋯ |
| L(s) = 1 | + (0.324 − 0.386i)3-s + (0.803 − 0.292i)5-s + (−0.734 − 1.27i)7-s + (0.129 + 0.733i)9-s + (0.455 − 0.789i)11-s + (−0.108 − 0.129i)13-s + (0.147 − 0.405i)15-s + (0.229 − 1.30i)17-s + (−0.0564 − 0.998i)19-s + (−0.730 − 0.128i)21-s + (0.426 + 0.155i)23-s + (−0.205 + 0.172i)25-s + (0.763 + 0.440i)27-s + (1.39 − 0.245i)29-s + (−0.839 + 0.484i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.45750 - 1.18213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45750 - 1.18213i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (20.3 + 360. i)T \) |
| good | 3 | \( 1 + (-2.92 + 3.48i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (-20.0 + 7.31i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (36.0 + 62.3i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-55.1 + 95.5i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (18.3 + 21.9i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-66.2 + 375. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-225. - 82.0i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-1.17e3 + 206. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (806. - 465. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.39e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.66e3 - 1.98e3i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-2.03e3 + 741. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-593. - 3.36e3i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (799. - 2.19e3i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (186. + 32.9i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (1.41e3 + 514. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-7.40e3 + 1.30e3i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (1.33e3 + 3.67e3i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-2.64e3 - 2.21e3i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-685. + 816. i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (2.62e3 + 4.54e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (333. + 397. i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-1.37e4 - 2.43e3i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−13.75337858068255449439550558588, −12.82590792638026482928303455131, −11.20936997098993988449712506568, −10.09056379341844915695757785431, −9.063369783810810481543991597958, −7.56907964767355380579930184650, −6.52614617217289130202847128526, −4.87046208901512947877284369340, −2.98244847747023077229706377604, −0.997248551433172573537428783225,
2.14639400919099157902392416536, 3.74271305065169205998121367021, 5.72146952693082803021835394650, 6.65448308585356960485006712627, 8.623733817347862812370773393743, 9.543674933760767031044855895728, 10.29350101231828004284943882817, 12.14450215351057546483123169301, 12.68986685799090187471300873816, 14.28345822053073694704165582754
