| L(s) = 1 | + (13.5 − 8.55i)2-s + (144. + 47.0i)3-s + (109. − 231. i)4-s + (328. − 531. i)5-s + (2.36e3 − 603. i)6-s − 844. i·7-s + (−498. − 4.06e3i)8-s + (1.34e4 + 9.79e3i)9-s + (−101. − 9.99e3i)10-s + (−7.82e3 − 1.07e4i)11-s + (2.67e4 − 2.83e4i)12-s + (1.46e4 + 1.06e4i)13-s + (−7.22e3 − 1.14e4i)14-s + (7.26e4 − 6.15e4i)15-s + (−4.15e4 − 5.07e4i)16-s + (3.92e4 + 1.20e5i)17-s + ⋯ |
| L(s) = 1 | + (0.844 − 0.534i)2-s + (1.78 + 0.581i)3-s + (0.428 − 0.903i)4-s + (0.526 − 0.850i)5-s + (1.82 − 0.465i)6-s − 0.351i·7-s + (−0.121 − 0.992i)8-s + (2.05 + 1.49i)9-s + (−0.0101 − 0.999i)10-s + (−0.534 − 0.735i)11-s + (1.29 − 1.36i)12-s + (0.511 + 0.371i)13-s + (−0.188 − 0.297i)14-s + (1.43 − 1.21i)15-s + (−0.633 − 0.773i)16-s + (0.470 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.96220 - 3.69909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.96220 - 3.69909i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-13.5 + 8.55i)T \) |
| 5 | \( 1 + (-328. + 531. i)T \) |
| good | 3 | \( 1 + (-144. - 47.0i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 844. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (7.82e3 + 1.07e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-1.46e4 - 1.06e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-3.92e4 - 1.20e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (5.01e4 - 1.62e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (2.96e5 + 4.07e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (1.91e5 - 5.88e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-9.60e4 + 3.11e4i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (7.26e5 + 5.28e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-4.08e6 - 2.96e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.15e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.48e5 - 8.06e4i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.75e3 + 5.40e3i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-6.75e6 + 9.29e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (5.81e6 - 4.22e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.79e7 + 9.06e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (3.26e7 + 1.06e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (2.52e7 - 1.83e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.44e7 + 7.93e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-2.29e7 + 7.47e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (3.57e7 - 2.59e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.40e7 - 4.31e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.67321495651528454507302164076, −10.71975290886690074844453895416, −10.00181135521920914459704713836, −8.866502848986288446515294836819, −8.032739105956481710657679084419, −6.08376280489078356843279940890, −4.53015781505635750018468063430, −3.72502534767037540471896951824, −2.44422613841579368127996036611, −1.35209866982219646560025328567,
2.04028816236272678466590345130, 2.76992150177008096944247900656, 3.84335560830060083446117031773, 5.71934889044877526922885029084, 7.17743738193141405060795656807, 7.64027370810964749623868598689, 8.936965420211536228172170462650, 10.04633535057285108317246580014, 11.83888757164491168445893222129, 12.97379686143555704215390122057
