| L(s) = 1 | + (336. − 194. i)2-s + (1.67e4 + 1.03e4i)3-s + (−5.53e4 + 9.59e4i)4-s + (2.58e6 + 1.49e6i)5-s + (7.65e6 + 2.11e5i)6-s + (−1.88e7 − 3.26e7i)7-s + 1.45e8i·8-s + (1.74e8 + 3.45e8i)9-s + 1.15e9·10-s + (−2.85e9 + 1.64e9i)11-s + (−1.91e9 + 1.03e9i)12-s + (1.41e9 − 2.45e9i)13-s + (−1.26e10 − 7.32e9i)14-s + (2.79e10 + 5.15e10i)15-s + (1.37e10 + 2.37e10i)16-s − 3.48e10i·17-s + ⋯ |
| L(s) = 1 | + (0.658 − 0.379i)2-s + (0.851 + 0.523i)3-s + (−0.211 + 0.365i)4-s + (1.32 + 0.762i)5-s + (0.759 + 0.0209i)6-s + (−0.466 − 0.807i)7-s + 1.08i·8-s + (0.451 + 0.892i)9-s + 1.15·10-s + (−1.21 + 0.698i)11-s + (−0.371 + 0.201i)12-s + (0.133 − 0.231i)13-s + (−0.614 − 0.354i)14-s + (0.726 + 1.34i)15-s + (0.199 + 0.345i)16-s − 0.293i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(3.06307 + 2.02112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.06307 + 2.02112i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.67e4 - 1.03e4i)T \) |
| good | 2 | \( 1 + (-336. + 194. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (-2.58e6 - 1.49e6i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (1.88e7 + 3.26e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (2.85e9 - 1.64e9i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (-1.41e9 + 2.45e9i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 + 3.48e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 - 1.82e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-2.70e12 - 1.56e12i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (5.66e12 - 3.26e12i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-1.57e13 + 2.72e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 - 1.46e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + (9.47e13 + 5.47e13i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (4.63e14 + 8.02e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (1.00e14 - 5.81e13i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 - 3.21e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (-7.72e15 - 4.46e15i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (5.26e15 + 9.12e15i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (-2.33e16 + 4.04e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 + 5.73e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 6.01e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (-7.20e16 - 1.24e17i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (-2.87e17 + 1.66e17i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 - 5.02e15iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-1.27e17 - 2.20e17i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−17.02855901173420669143471453143, −15.09567209366912226221377776335, −13.64716428467403059889288793840, −13.24707664824507323036992527670, −10.65833508456516458798011664210, −9.538605247678194164608877748023, −7.47471508786110948043213612517, −5.11193417351226732257965386576, −3.33794641828487244374674149375, −2.31332733212988070641779130662,
1.07807311884680632758775351986, 2.76791888466520971458928008202, 5.17143598712144514042110723645, 6.34189261892132145880557266925, 8.701783260935052630334439939938, 9.818985810623390217962460096466, 12.91214810075022172586929555337, 13.34468802101135566367748859478, 14.68384050458645177390121716134, 16.10966713655485236333970407943
