L(s) = 1 | + (0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (−2.52 − 0.781i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−1.31 − 2.27i)11-s + (−0.965 + 0.258i)12-s + (−1.21 + 1.21i)13-s + (0.101 − 2.64i)14-s + (0.500 − 0.866i)16-s + (−1.95 + 7.31i)17-s + (−0.258 + 0.965i)18-s + (−2.32 + 4.02i)19-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (−0.955 − 0.295i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.395 − 0.685i)11-s + (−0.278 + 0.0747i)12-s + (−0.337 + 0.337i)13-s + (0.0270 − 0.706i)14-s + (0.125 − 0.216i)16-s + (−0.475 + 1.77i)17-s + (−0.0610 + 0.227i)18-s + (−0.533 + 0.924i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7687452502\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7687452502\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.52 + 0.781i)T \) |
good | 11 | \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.21 - 1.21i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.95 - 7.31i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.32 - 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.95 - 1.32i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.99iT - 29T^{2} \) |
| 31 | \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.02 + 3.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.59iT - 41T^{2} \) |
| 43 | \( 1 + (-0.545 - 0.545i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.12 + 1.64i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.22 + 8.28i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.86 + 6.68i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.16 - 2.40i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.47 + 0.663i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 + (-13.1 - 3.53i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.78 - 4.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.99 + 7.99i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.0812 - 0.140i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.35 - 4.35i)T + 97iT^{2} \) |
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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
−10.35988504125457356610443494093, −9.287389644194625558628113288366, −8.639472113611547855445735965631, −7.88580910253977407554982303683, −6.96188322223746868518452904730, −6.16867076170527090262036306639, −5.35641935573627972573302605756, −3.88564800202627719734752307311, −3.59142293300102641406970418703, −2.00998384621524033289570325489,
0.27878805812518909116052307650, 2.30564646735008526844970909663, 2.76414312045829987727362706418, 4.02020401599819001298619720743, 4.91580661648568834223908850291, 6.03498839980518054232493545147, 7.05731460414519499867985149705, 7.81439961160535780005843847404, 9.023141321021712916887379912702, 9.489555121585873672894802251548