| L(s) = 1 | + (3.28 − 6.82i)2-s + (−6.60 − 5.27i)3-s + (−15.8 − 19.8i)4-s + (−62.0 − 29.8i)5-s + (−57.7 + 27.7i)6-s + (−19.9 + 24.9i)7-s + (48.4 − 11.0i)8-s + (−38.1 − 167. i)9-s + (−408. + 325. i)10-s + (65.7 + 14.9i)11-s + 215. i·12-s + (−9.71 + 42.5i)13-s + (105. + 218. i)14-s + (252. + 524. i)15-s + (264. − 1.16e3i)16-s − 926. i·17-s + ⋯ |
| L(s) = 1 | + (0.581 − 1.20i)2-s + (−0.423 − 0.338i)3-s + (−0.495 − 0.621i)4-s + (−1.11 − 0.534i)5-s + (−0.654 + 0.315i)6-s + (−0.153 + 0.192i)7-s + (0.267 − 0.0610i)8-s + (−0.157 − 0.688i)9-s + (−1.29 + 1.02i)10-s + (0.163 + 0.0373i)11-s + 0.431i·12-s + (−0.0159 + 0.0698i)13-s + (0.143 + 0.297i)14-s + (0.289 + 0.602i)15-s + (0.258 − 1.13i)16-s − 0.777i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0778i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0515231 - 1.32237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0515231 - 1.32237i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-4.48e3 - 624. i)T \) |
| good | 2 | \( 1 + (-3.28 + 6.82i)T + (-19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (6.60 + 5.27i)T + (54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (62.0 + 29.8i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (19.9 - 24.9i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-65.7 - 14.9i)T + (1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (9.71 - 42.5i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 926. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.06e3 + 848. i)T + (5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-2.46e3 + 1.18e3i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (3.23e3 - 6.71e3i)T + (-1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (7.86e3 - 1.79e3i)T + (6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 1.11e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (3.16e3 + 6.57e3i)T + (-9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-9.20e3 - 2.10e3i)T + (2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.79e4 - 1.34e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + 1.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-2.18e3 - 1.74e3i)T + (1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (1.12e4 + 4.92e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.74e4 + 7.64e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (7.03e3 + 1.46e4i)T + (-1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (-5.94e4 + 1.35e4i)T + (2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (-1.89e4 - 2.37e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (4.41e4 - 9.15e4i)T + (-3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-8.49e4 + 6.77e4i)T + (1.91e9 - 8.37e9i)T^{2} \) |
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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
−15.46326962922470914635990325076, −13.82424922398924089306342424326, −12.32151027469240486364750503224, −12.06133721254415796401530566321, −10.88283614192457245658817316305, −9.046528833375198265233359203076, −7.14135711623949275618455628307, −4.86117732797818116309661226398, −3.28147395989450764256406268312, −0.78716825636749267205038783122,
3.95902149433713113805123308181, 5.49130601067425612294213265247, 7.05322345247675707120267564635, 8.109459082752010788193625856404, 10.47038016836451876315107821482, 11.59802592577905211998855704389, 13.32683309509237167620492946776, 14.60071458418927942768514958861, 15.50497163094376554640582626534, 16.34340077816668333108020837541
