| L(s) = 1 | + (−1.33 + 0.467i)2-s + (0.411 + 3.64i)3-s + (1.56 − 1.24i)4-s + (0.218 − 0.454i)5-s + (−2.25 − 4.67i)6-s + (−7.64 + 9.59i)7-s + (−1.50 + 2.39i)8-s + (−4.37 + 0.997i)9-s + (−0.0799 + 0.709i)10-s + (0.992 + 1.57i)11-s + (5.19 + 5.19i)12-s + (10.1 + 2.30i)13-s + (5.72 − 16.3i)14-s + (1.74 + 0.612i)15-s + (0.890 − 3.89i)16-s + (14.5 − 14.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.667 + 0.233i)2-s + (0.137 + 1.21i)3-s + (0.390 − 0.311i)4-s + (0.0437 − 0.0909i)5-s + (−0.375 − 0.779i)6-s + (−1.09 + 1.37i)7-s + (−0.188 + 0.299i)8-s + (−0.485 + 0.110i)9-s + (−0.00799 + 0.0709i)10-s + (0.0901 + 0.143i)11-s + (0.432 + 0.432i)12-s + (0.777 + 0.177i)13-s + (0.409 − 1.16i)14-s + (0.116 + 0.0408i)15-s + (0.0556 − 0.243i)16-s + (0.858 − 0.858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.474062 + 0.697837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474062 + 0.697837i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.467i)T \) |
| 29 | \( 1 + (-4.39 + 28.6i)T \) |
| good | 3 | \( 1 + (-0.411 - 3.64i)T + (-8.77 + 2.00i)T^{2} \) |
| 5 | \( 1 + (-0.218 + 0.454i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (7.64 - 9.59i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (-0.992 - 1.57i)T + (-52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (-10.1 - 2.30i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (-14.5 + 14.5i)T - 289iT^{2} \) |
| 19 | \( 1 + (9.66 + 1.08i)T + (351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (-31.2 + 15.0i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (-3.66 + 1.28i)T + (751. - 599. i)T^{2} \) |
| 37 | \( 1 + (28.8 - 45.9i)T + (-593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (19.7 + 19.7i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-13.9 + 39.8i)T + (-1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (55.8 - 35.0i)T + (958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (41.8 + 20.1i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 69.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-3.94 - 35.0i)T + (-3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (-111. + 25.4i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (44.2 + 10.0i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (7.08 + 2.47i)T + (4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (119. + 74.8i)T + (2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 12.9i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-141. + 49.5i)T + (6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-2.65 + 23.5i)T + (-9.17e3 - 2.09e3i)T^{2} \) |
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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
−15.55901824864154307741345757271, −14.71671184441399882523784226598, −12.96770708547880640656692139746, −11.64069598626368646657282278098, −10.27608566120418118659523343981, −9.353297129068007633300719794662, −8.685022657642728390760700366680, −6.62275860047692305763029445488, −5.19814969627995067249747304040, −3.13298060734985086740534238111,
1.10524545315288457574800217237, 3.43002348639116423745702933750, 6.44435638616473238941529519988, 7.22652889506560312484917960739, 8.446593444369151041667007959328, 10.01671042063103914468371530350, 10.97548056645870480950310121140, 12.65895949092696069094438434789, 13.10886073031330051507393272849, 14.30919901653848782195143251047
