| L(s) = 1 | + (28.1 + 6.42i)2-s + (−39.5 − 82.1i)3-s + (289. + 139. i)4-s + (−3.26 + 14.3i)5-s + (−585. − 2.56e3i)6-s + (2.32e3 − 1.12e3i)7-s + (−4.30e3 − 3.43e3i)8-s + (7.08e3 − 8.88e3i)9-s + (−183. + 381. i)10-s + (3.58e4 − 2.85e4i)11-s − 2.93e4i·12-s + (−1.04e5 − 1.31e5i)13-s + (7.26e4 − 1.65e4i)14-s + (1.30e3 − 297. i)15-s + (−2.01e5 − 2.52e5i)16-s − 2.11e5i·17-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.283i)2-s + (−0.282 − 0.585i)3-s + (0.565 + 0.272i)4-s + (−0.00233 + 0.0102i)5-s + (−0.184 − 0.808i)6-s + (0.366 − 0.176i)7-s + (−0.371 − 0.296i)8-s + (0.359 − 0.451i)9-s + (−0.00581 + 0.0120i)10-s + (0.738 − 0.588i)11-s − 0.407i·12-s + (−1.01 − 1.27i)13-s + (0.505 − 0.115i)14-s + (0.00665 − 0.00151i)15-s + (−0.769 − 0.964i)16-s − 0.612i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.18854 - 1.84260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18854 - 1.84260i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-2.77e6 + 2.60e6i)T \) |
| good | 2 | \( 1 + (-28.1 - 6.42i)T + (461. + 222. i)T^{2} \) |
| 3 | \( 1 + (39.5 + 82.1i)T + (-1.22e4 + 1.53e4i)T^{2} \) |
| 5 | \( 1 + (3.26 - 14.3i)T + (-1.75e6 - 8.47e5i)T^{2} \) |
| 7 | \( 1 + (-2.32e3 + 1.12e3i)T + (2.51e7 - 3.15e7i)T^{2} \) |
| 11 | \( 1 + (-3.58e4 + 2.85e4i)T + (5.24e8 - 2.29e9i)T^{2} \) |
| 13 | \( 1 + (1.04e5 + 1.31e5i)T + (-2.35e9 + 1.03e10i)T^{2} \) |
| 17 | \( 1 + 2.11e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + (5.62e4 - 1.16e5i)T + (-2.01e11 - 2.52e11i)T^{2} \) |
| 23 | \( 1 + (-4.94e5 - 2.16e6i)T + (-1.62e12 + 7.81e11i)T^{2} \) |
| 31 | \( 1 + (-7.80e5 - 1.78e5i)T + (2.38e13 + 1.14e13i)T^{2} \) |
| 37 | \( 1 + (-2.42e6 - 1.93e6i)T + (2.89e13 + 1.26e14i)T^{2} \) |
| 41 | \( 1 - 2.45e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-7.04e6 + 1.60e6i)T + (4.52e14 - 2.18e14i)T^{2} \) |
| 47 | \( 1 + (5.05e4 - 4.03e4i)T + (2.49e14 - 1.09e15i)T^{2} \) |
| 53 | \( 1 + (-1.82e6 + 8.00e6i)T + (-2.97e15 - 1.43e15i)T^{2} \) |
| 59 | \( 1 + 1.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (1.75e7 + 3.64e7i)T + (-7.29e15 + 9.14e15i)T^{2} \) |
| 67 | \( 1 + (-1.56e8 + 1.96e8i)T + (-6.05e15 - 2.65e16i)T^{2} \) |
| 71 | \( 1 + (6.80e7 + 8.52e7i)T + (-1.02e16 + 4.46e16i)T^{2} \) |
| 73 | \( 1 + (1.07e8 - 2.45e7i)T + (5.30e16 - 2.55e16i)T^{2} \) |
| 79 | \( 1 + (6.37e7 + 5.08e7i)T + (2.66e16 + 1.16e17i)T^{2} \) |
| 83 | \( 1 + (-6.77e8 - 3.26e8i)T + (1.16e17 + 1.46e17i)T^{2} \) |
| 89 | \( 1 + (-4.54e8 - 1.03e8i)T + (3.15e17 + 1.52e17i)T^{2} \) |
| 97 | \( 1 + (-6.10e8 + 1.26e9i)T + (-4.73e17 - 5.94e17i)T^{2} \) |
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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
−14.63429461331532884702426592053, −13.48974742758438527668402822221, −12.52399882238595850281900224256, −11.52871178273305394223576931580, −9.575775989412533956750713172492, −7.51390763816402977076748620031, −6.23056856978784901894092830959, −4.92126078319933928950604374115, −3.26378799614407913825578466246, −0.836824010474792921312724015232,
2.20135926459415162930464215540, 4.24501667878220710549604286962, 4.91041289470822043739810898016, 6.72959541930103135656515351668, 8.916254254617197504959376926346, 10.51850459057071261050177926991, 11.83368269521020373690277159520, 12.72280693780294306530804327735, 14.26794026241618399786143467532, 14.85797302151030698256126500603
