| L(s) = 1 | + (−1.08 + 2.24i)2-s + (−24.6 − 51.1i)3-s + (155. + 195. i)4-s + (−234. + 53.6i)5-s + 141.·6-s + 271. i·7-s + (−1.22e3 + 280. i)8-s + (2.07e3 − 2.60e3i)9-s + (133. − 585. i)10-s + (−8.44e3 + 1.05e4i)11-s + (6.15e3 − 1.27e4i)12-s + (−5.83e3 − 2.55e4i)13-s + (−610. − 293. i)14-s + (8.53e3 + 1.07e4i)15-s + (−1.35e4 + 5.92e4i)16-s + (4.85e3 − 2.12e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0675 + 0.140i)2-s + (−0.304 − 0.631i)3-s + (0.608 + 0.762i)4-s + (−0.375 + 0.0858i)5-s + 0.109·6-s + 0.113i·7-s + (−0.300 + 0.0684i)8-s + (0.316 − 0.397i)9-s + (0.0133 − 0.0585i)10-s + (−0.576 + 0.723i)11-s + (0.296 − 0.616i)12-s + (−0.204 − 0.894i)13-s + (−0.0158 − 0.00764i)14-s + (0.168 + 0.211i)15-s + (−0.206 + 0.904i)16-s + (0.0581 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0565926 + 0.369219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0565926 + 0.369219i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.30e6 - 2.52e6i)T \) |
| good | 2 | \( 1 + (1.08 - 2.24i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (24.6 + 51.1i)T + (-4.09e3 + 5.12e3i)T^{2} \) |
| 5 | \( 1 + (234. - 53.6i)T + (3.51e5 - 1.69e5i)T^{2} \) |
| 7 | \( 1 - 271. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (8.44e3 - 1.05e4i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (5.83e3 + 2.55e4i)T + (-7.34e8 + 3.53e8i)T^{2} \) |
| 17 | \( 1 + (-4.85e3 + 2.12e4i)T + (-6.28e9 - 3.02e9i)T^{2} \) |
| 19 | \( 1 + (1.39e5 - 1.11e5i)T + (3.77e9 - 1.65e10i)T^{2} \) |
| 23 | \( 1 + (2.57e5 - 3.23e5i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 29 | \( 1 + (-2.66e5 + 5.52e5i)T + (-3.11e11 - 3.91e11i)T^{2} \) |
| 31 | \( 1 + (1.48e6 + 7.17e5i)T + (5.31e11 + 6.66e11i)T^{2} \) |
| 37 | \( 1 - 2.79e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (2.88e6 + 1.38e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-3.23e6 - 4.05e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-2.29e6 + 1.00e7i)T + (-5.60e13 - 2.70e13i)T^{2} \) |
| 59 | \( 1 + (-3.62e6 + 1.58e7i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (8.22e5 + 1.70e6i)T + (-1.19e14 + 1.49e14i)T^{2} \) |
| 67 | \( 1 + (-6.93e6 - 8.69e6i)T + (-9.03e13 + 3.95e14i)T^{2} \) |
| 71 | \( 1 + (3.08e6 - 2.45e6i)T + (1.43e14 - 6.29e14i)T^{2} \) |
| 73 | \( 1 + (4.32e7 - 9.86e6i)T + (7.26e14 - 3.49e14i)T^{2} \) |
| 79 | \( 1 - 6.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.31e7 - 1.59e7i)T + (1.40e15 - 1.76e15i)T^{2} \) |
| 89 | \( 1 + (3.76e7 + 7.81e7i)T + (-2.45e15 + 3.07e15i)T^{2} \) |
| 97 | \( 1 + (2.48e7 - 3.11e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−15.09014305832642018340260403033, −13.15015909409267123328130755696, −12.37836610177818681818503010187, −11.49677351926477151376697622106, −9.953786521565541958910318744611, −8.024516487276438673396026229532, −7.31339979585365427795472024721, −5.93741608032913709024078223020, −3.76526648784944297526694696942, −2.01893646249480641687801108852,
0.13651143680054979740092210446, 2.13517593234758012400023884323, 4.24487951027500211770254654493, 5.65184305677055849588351271376, 7.12774490013465451242322199893, 8.877191595086064582370117816810, 10.43231523613889074289457887608, 10.89615927100255719069192181071, 12.24007501387615706922745743217, 13.84590159075899898565381513113
