| L(s) = 1 | + (−26.6 − 36.6i)2-s + (−123. − 378. i)3-s + (−632. + 1.94e3i)4-s + (9.05e3 + 6.57e3i)5-s + (−1.05e4 + 1.45e4i)6-s + (−7.37e4 − 2.39e4i)7-s + (8.81e4 − 2.86e4i)8-s + (3.01e5 − 2.19e5i)9-s − 5.06e5i·10-s + (1.75e6 − 2.66e5i)11-s + 8.15e5·12-s + (−3.42e6 − 4.71e6i)13-s + (1.08e6 + 3.33e6i)14-s + (1.37e6 − 4.23e6i)15-s + (−3.39e6 − 2.46e6i)16-s + (−1.03e7 + 1.42e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.168 − 0.519i)3-s + (−0.154 + 0.475i)4-s + (0.579 + 0.420i)5-s + (−0.227 + 0.312i)6-s + (−0.626 − 0.203i)7-s + (0.336 − 0.109i)8-s + (0.567 − 0.412i)9-s − 0.506i·10-s + (0.988 − 0.150i)11-s + 0.273·12-s + (−0.709 − 0.977i)13-s + (0.144 + 0.443i)14-s + (0.120 − 0.372i)15-s + (−0.202 − 0.146i)16-s + (−0.429 + 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.103710 - 0.936926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103710 - 0.936926i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (26.6 + 36.6i)T \) |
| 11 | \( 1 + (-1.75e6 + 2.66e5i)T \) |
| good | 3 | \( 1 + (123. + 378. i)T + (-4.29e5 + 3.12e5i)T^{2} \) |
| 5 | \( 1 + (-9.05e3 - 6.57e3i)T + (7.54e7 + 2.32e8i)T^{2} \) |
| 7 | \( 1 + (7.37e4 + 2.39e4i)T + (1.11e10 + 8.13e9i)T^{2} \) |
| 13 | \( 1 + (3.42e6 + 4.71e6i)T + (-7.19e12 + 2.21e13i)T^{2} \) |
| 17 | \( 1 + (1.03e7 - 1.42e7i)T + (-1.80e14 - 5.54e14i)T^{2} \) |
| 19 | \( 1 + (-4.87e7 + 1.58e7i)T + (1.79e15 - 1.30e15i)T^{2} \) |
| 23 | \( 1 + 2.84e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (3.19e8 + 1.03e8i)T + (2.86e17 + 2.07e17i)T^{2} \) |
| 31 | \( 1 + (1.05e9 - 7.67e8i)T + (2.43e17 - 7.49e17i)T^{2} \) |
| 37 | \( 1 + (-6.93e8 + 2.13e9i)T + (-5.32e18 - 3.86e18i)T^{2} \) |
| 41 | \( 1 + (-3.58e9 + 1.16e9i)T + (1.82e19 - 1.32e19i)T^{2} \) |
| 43 | \( 1 + 6.42e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-1.96e9 - 6.03e9i)T + (-9.40e19 + 6.82e19i)T^{2} \) |
| 53 | \( 1 + (2.43e9 - 1.76e9i)T + (1.51e20 - 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-1.02e10 + 3.16e10i)T + (-1.43e21 - 1.04e21i)T^{2} \) |
| 61 | \( 1 + (-3.79e10 + 5.21e10i)T + (-8.20e20 - 2.52e21i)T^{2} \) |
| 67 | \( 1 + 9.58e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (1.15e11 + 8.38e10i)T + (5.07e21 + 1.56e22i)T^{2} \) |
| 73 | \( 1 + (1.78e11 + 5.81e10i)T + (1.85e22 + 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-8.96e10 - 1.23e11i)T + (-1.82e22 + 5.61e22i)T^{2} \) |
| 83 | \( 1 + (-1.12e11 + 1.54e11i)T + (-3.30e22 - 1.01e23i)T^{2} \) |
| 89 | \( 1 - 4.58e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-7.39e11 + 5.37e11i)T + (2.14e23 - 6.59e23i)T^{2} \) |
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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
−14.29324981128371536571234096556, −12.97706222781421204400060910722, −11.99180199558717148884283666187, −10.35264658835666864963376744358, −9.392558845174566571687111359450, −7.45434231563374729561781936398, −6.13992302096408218566459089554, −3.67362960448225731921028688965, −1.93939262744591314400433962909, −0.39409103306940044250523869692,
1.70969715452632590460335838244, 4.29978930339450864739330038924, 5.81732131364397378526738483286, 7.30917384757014590049326716179, 9.345801495493542745425439528169, 9.788876203151177743621224282286, 11.71438023501267870445640574463, 13.36944340428732948048171983610, 14.61811726305258272752428283187, 16.15384535184755490558260566213
