| L(s) = 1 | + (0.618 − 1.90i)2-s + (−6.91 + 5.02i)3-s + (−3.23 − 2.35i)4-s + (3.93 + 12.1i)5-s + (5.28 + 16.2i)6-s + (18.9 + 13.7i)7-s + (−6.47 + 4.70i)8-s + (14.2 − 43.7i)9-s + 25.4·10-s + 34.1·12-s + (−3.54 + 10.9i)13-s + (37.9 − 27.5i)14-s + (−88.0 − 63.9i)15-s + (4.94 + 15.2i)16-s + (20.2 + 62.2i)17-s + (−74.4 − 54.0i)18-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−1.33 + 0.966i)3-s + (−0.404 − 0.293i)4-s + (0.352 + 1.08i)5-s + (0.359 + 1.10i)6-s + (1.02 + 0.744i)7-s + (−0.286 + 0.207i)8-s + (0.526 − 1.62i)9-s + 0.805·10-s + 0.822·12-s + (−0.0755 + 0.232i)13-s + (0.724 − 0.526i)14-s + (−1.51 − 1.10i)15-s + (0.0772 + 0.237i)16-s + (0.288 + 0.888i)17-s + (−0.974 − 0.708i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.302306 + 0.860259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.302306 + 0.860259i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 + 1.90i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (6.91 - 5.02i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-3.93 - 12.1i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-18.9 - 13.7i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (3.54 - 10.9i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-20.2 - 62.2i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (5.86 - 4.26i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (103. + 74.8i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (89.2 - 274. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-69.0 - 50.1i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (109. - 79.6i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-108. + 79.1i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-154. + 476. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (527. + 383. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (113. + 347. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-40.9 - 125. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-379. - 275. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (126. - 388. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-420. - 1.29e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 260.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-437. + 1.34e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.71258163413280132875750320072, −11.08797783404296803605130636696, −10.41620041369704554063415062037, −9.697514799234414827561061548261, −8.332733406933210911391075310299, −6.57926706042892799054480730281, −5.66309556506474161231392916813, −4.82744191335672235069828927653, −3.57449299203843224943581635594, −1.89539511396869767546540605429,
0.41791514463300017327639473871, 1.56469574483998749441246288671, 4.43126348357017610141134618882, 5.26332840115373133723422649670, 6.00957341983218037317094552367, 7.32160835964252340829876427691, 7.85641435740612312964908655685, 9.179537927419253260471178560860, 10.55715456498280155105052860750, 11.59856334713664447391585448480
