| L(s) = 1 | + (30.7 + 5.42i)3-s + (−106. − 89.1i)5-s + (−115. − 200. i)7-s + (231. + 84.2i)9-s + (−1.12e3 + 1.94e3i)11-s + (−4.19e3 + 739. i)13-s + (−2.78e3 − 3.31e3i)15-s + (3.10e3 − 1.13e3i)17-s + (4.41e3 + 5.24e3i)19-s + (−2.47e3 − 6.79e3i)21-s + (−5.69e3 + 4.77e3i)23-s + (627. + 3.55e3i)25-s + (−1.30e4 − 7.53e3i)27-s + (1.41e4 − 3.87e4i)29-s + (−3.99e4 + 2.30e4i)31-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.200i)3-s + (−0.850 − 0.713i)5-s + (−0.337 − 0.584i)7-s + (0.317 + 0.115i)9-s + (−0.842 + 1.45i)11-s + (−1.90 + 0.336i)13-s + (−0.824 − 0.983i)15-s + (0.632 − 0.230i)17-s + (0.644 + 0.764i)19-s + (−0.267 − 0.733i)21-s + (−0.468 + 0.392i)23-s + (0.0401 + 0.227i)25-s + (−0.663 − 0.382i)27-s + (0.578 − 1.58i)29-s + (−1.33 + 0.773i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0520i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.998 - 0.0520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.00189751 + 0.0728142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00189751 + 0.0728142i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.41e3 - 5.24e3i)T \) |
| good | 3 | \( 1 + (-30.7 - 5.42i)T + (685. + 249. i)T^{2} \) |
| 5 | \( 1 + (106. + 89.1i)T + (2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (115. + 200. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.12e3 - 1.94e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (4.19e3 - 739. i)T + (4.53e6 - 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-3.10e3 + 1.13e3i)T + (1.84e7 - 1.55e7i)T^{2} \) |
| 23 | \( 1 + (5.69e3 - 4.77e3i)T + (2.57e7 - 1.45e8i)T^{2} \) |
| 29 | \( 1 + (-1.41e4 + 3.87e4i)T + (-4.55e8 - 3.82e8i)T^{2} \) |
| 31 | \( 1 + (3.99e4 - 2.30e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 6.29e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (6.09e4 + 1.07e4i)T + (4.46e9 + 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-1.05e5 - 8.84e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (7.20e4 + 2.62e4i)T + (8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 + (-2.11e4 - 2.52e4i)T + (-3.84e9 + 2.18e10i)T^{2} \) |
| 59 | \( 1 + (5.86e4 + 1.61e5i)T + (-3.23e10 + 2.71e10i)T^{2} \) |
| 61 | \( 1 + (7.33e4 - 6.15e4i)T + (8.94e9 - 5.07e10i)T^{2} \) |
| 67 | \( 1 + (-1.44e5 + 3.97e5i)T + (-6.92e10 - 5.81e10i)T^{2} \) |
| 71 | \( 1 + (2.09e5 - 2.50e5i)T + (-2.22e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-1.95e4 + 1.11e5i)T + (-1.42e11 - 5.17e10i)T^{2} \) |
| 79 | \( 1 + (-5.91e5 - 1.04e5i)T + (2.28e11 + 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-1.59e5 - 2.76e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (4.88e5 - 8.61e4i)T + (4.67e11 - 1.69e11i)T^{2} \) |
| 97 | \( 1 + (-4.74e5 - 1.30e6i)T + (-6.38e11 + 5.35e11i)T^{2} \) |
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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
−12.63483825827078602608549387139, −12.00906965218636041145799023802, −10.03496463911719968419303420262, −9.453297766028219341675997364972, −7.83871343432919968468777946009, −7.46864373026010920614716188835, −4.96830882032511562837223875848, −3.83158015436647312538822987730, −2.29165803203738570204801069661, −0.02188906976128796559209727922,
2.68422562054247018994352184639, 3.26118719005614413157978284025, 5.40078637829718634948019973341, 7.25284783846383735825261480666, 8.029132629110990290651330742714, 9.127947548503670076784799875488, 10.48958519373786926975145014413, 11.72275003060494800712663011629, 12.85024970533101512892680194622, 14.05125855099294808811725541179
