L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (3.31 − 3.74i)5-s + (0.704 + 0.406i)7-s − 2.82·8-s + (−2.24 − 6.70i)10-s + (−13.3 − 7.68i)11-s + (−5.10 + 2.94i)13-s + (0.996 − 0.575i)14-s + (−2.00 + 3.46i)16-s − 12.8·17-s + 1.24·19-s + (−9.79 − 1.99i)20-s + (−18.8 + 10.8i)22-s + (2.39 + 4.15i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.662 − 0.748i)5-s + (0.100 + 0.0581i)7-s − 0.353·8-s + (−0.224 − 0.670i)10-s + (−1.21 − 0.698i)11-s + (−0.392 + 0.226i)13-s + (0.0711 − 0.0410i)14-s + (−0.125 + 0.216i)16-s − 0.758·17-s + 0.0654·19-s + (−0.489 − 0.0997i)20-s + (−0.855 + 0.494i)22-s + (0.104 + 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.047064132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047064132\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.31 + 3.74i)T \) |
good | 7 | \( 1 + (-0.704 - 0.406i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (13.3 + 7.68i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.10 - 2.94i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.8T + 289T^{2} \) |
| 19 | \( 1 - 1.24T + 361T^{2} \) |
| 23 | \( 1 + (-2.39 - 4.15i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (36.9 + 21.3i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (2.10 + 3.64i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 70.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-6.09 + 3.52i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (35.5 + 20.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-39.8 + 69.1i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 63.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (31.7 - 18.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-41.4 + 71.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (77.0 - 44.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 69.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 89.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (67.0 - 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (54.5 - 94.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (78.4 + 45.3i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.765495116943966444907509203409, −8.797536169127615290978025921770, −8.139633129792236423988145556417, −6.83965098024756968272827542006, −5.65342936061646373282582786533, −5.16157884552744510501618558804, −4.10804013910753723116328626251, −2.76688406007992074523015923868, −1.81074646093997913882376712388, −0.28349056199335134225424521840,
2.09138761141746423621700220638, 3.03548737831584495105297620840, 4.39623028454586611680212126169, 5.35825582067984182343999032342, 6.08073331269603452922736376154, 7.27560470720653643265265593444, 7.50608927319200846947326510141, 8.839374195868927835822493049328, 9.644719276300861632386803953870, 10.56278028648636027595409259857