| L(s) = 1 | + (1.74 − 1.11i)2-s + (1.28 − 8.90i)3-s + (−11.5 + 25.2i)4-s + (1.84 − 0.543i)5-s + (−7.74 − 16.9i)6-s + (−46.8 − 54.1i)7-s + (17.5 + 122. i)8-s + (−77.7 − 22.8i)9-s + (2.61 − 3.01i)10-s + (−573. − 368. i)11-s + (209. + 134. i)12-s + (−538. + 622. i)13-s + (−142. − 41.7i)14-s + (−2.46 − 17.1i)15-s + (−413. − 476. i)16-s + (99.4 + 217. i)17-s + ⋯ |
| L(s) = 1 | + (0.307 − 0.197i)2-s + (0.0821 − 0.571i)3-s + (−0.359 + 0.787i)4-s + (0.0330 − 0.00971i)5-s + (−0.0878 − 0.192i)6-s + (−0.361 − 0.417i)7-s + (0.0972 + 0.676i)8-s + (−0.319 − 0.0939i)9-s + (0.00826 − 0.00953i)10-s + (−1.42 − 0.917i)11-s + (0.420 + 0.270i)12-s + (−0.884 + 1.02i)13-s + (−0.193 − 0.0569i)14-s + (−0.00283 − 0.0197i)15-s + (−0.403 − 0.465i)16-s + (0.0834 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0307098 + 0.137053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0307098 + 0.137053i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.28 + 8.90i)T \) |
| 23 | \( 1 + (-1.96e3 - 1.60e3i)T \) |
| good | 2 | \( 1 + (-1.74 + 1.11i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (-1.84 + 0.543i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (46.8 + 54.1i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (573. + 368. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (538. - 622. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-99.4 - 217. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (683. - 1.49e3i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (1.26e3 + 2.77e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (1.49e3 + 1.04e4i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-6.80e3 - 1.99e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (1.42e4 - 4.17e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-793. + 5.52e3i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 - 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.31e4 - 2.66e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-112. + 130. i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (5.16e3 + 3.59e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-1.97e3 + 1.26e3i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (2.00e4 - 1.28e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (-7.19e3 + 1.57e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (5.45e4 - 6.29e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (-1.34e4 - 3.95e3i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (1.96e4 - 1.36e5i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (1.52e5 - 4.48e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−13.65750008312552213798436467745, −13.32931578515341403439148229337, −12.18936103771618242621440909562, −11.15935887812015966692251701884, −9.611983108755881076023232601014, −8.185715916412420735256775114944, −7.34555929292702953411580320847, −5.60838626715170107494328141497, −3.91425849649418708745065456370, −2.44247791709278874667726526643,
0.05273986557549743966858229345, 2.67466755786221998523893669584, 4.73251850423998502528981010303, 5.48776787174862360085610026010, 7.15984068924364747831998356282, 8.820242051709699009500637504120, 10.03838673083059461500543511100, 10.59152072187577134518378619062, 12.47390411619026802045932916703, 13.28634950156322025967153055178
