L(s) = 1 | + (1.60 − 2.53i)3-s + (−8.13 + 4.69i)7-s + (−3.82 − 8.14i)9-s + (15.4 − 8.90i)11-s + (11.9 + 6.92i)13-s + 30.7·17-s − 28.5·19-s + (−1.19 + 28.1i)21-s + (6.16 − 10.6i)23-s + (−26.7 − 3.42i)27-s + (5.02 − 2.90i)29-s + (−3.25 + 5.63i)31-s + (2.26 − 53.3i)33-s − 66.5i·37-s + (36.8 − 19.2i)39-s + ⋯ |
L(s) = 1 | + (0.536 − 0.844i)3-s + (−1.16 + 0.670i)7-s + (−0.424 − 0.905i)9-s + (1.40 − 0.809i)11-s + (0.922 + 0.532i)13-s + 1.80·17-s − 1.50·19-s + (−0.0568 + 1.34i)21-s + (0.267 − 0.464i)23-s + (−0.991 − 0.126i)27-s + (0.173 − 0.100i)29-s + (−0.104 + 0.181i)31-s + (0.0686 − 1.61i)33-s − 1.79i·37-s + (0.943 − 0.492i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0297 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0297 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.066954594\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066954594\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.60 + 2.53i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (8.13 - 4.69i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.4 + 8.90i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-11.9 - 6.92i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 30.7T + 289T^{2} \) |
| 19 | \( 1 + 28.5T + 361T^{2} \) |
| 23 | \( 1 + (-6.16 + 10.6i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 2.90i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (3.25 - 5.63i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 66.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (33.0 + 19.0i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-47.7 + 27.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (8.23 + 14.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 69.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (91.2 + 52.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.5 + 58.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.8 - 22.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-47.3 - 81.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-7.73 - 13.4i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 52.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (66.0 - 38.1i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.278091205694471318652443468043, −8.974851436844050157696668028515, −8.176408296543460007918625782988, −6.98348292008820824431211655777, −6.26791313289228749697855470953, −5.81240442121319994041027720441, −3.84724983260107607104093311883, −3.30449250190960204481099769378, −2.00429558468746977860635192328, −0.70264840254697542424549333643,
1.29478598879223699348507647632, 3.03777739596203819592947784607, 3.73117306814431835278416243702, 4.48790540760737836842705596655, 5.85043514448313088392917859656, 6.64439182316626499635102912592, 7.66855720677961566880069609628, 8.605457504438855582668479317738, 9.435116881173147700194194151191, 10.05584067604964565461609601797