| L(s) = 1 | + (−6.47 + 4.15i)2-s + (2.21 − 15.4i)3-s + (−2.01 + 4.40i)4-s + (−59.1 − 201. i)5-s + (49.8 + 109. i)6-s + (−482. + 418. i)7-s + (−75.3 − 524. i)8-s + (−233. − 68.4i)9-s + (1.22e3 + 1.05e3i)10-s + (−86.0 + 133. i)11-s + (63.5 + 40.8i)12-s + (1.17e3 − 1.35e3i)13-s + (1.38e3 − 4.71e3i)14-s + (−3.23e3 + 465. i)15-s + (2.46e3 + 2.84e3i)16-s + (173. − 79.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.808 + 0.519i)2-s + (0.0821 − 0.571i)3-s + (−0.0314 + 0.0688i)4-s + (−0.473 − 1.61i)5-s + (0.230 + 0.504i)6-s + (−1.40 + 1.21i)7-s + (−0.147 − 1.02i)8-s + (−0.319 − 0.0939i)9-s + (1.22 + 1.05i)10-s + (−0.0646 + 0.100i)11-s + (0.0367 + 0.0236i)12-s + (0.533 − 0.615i)13-s + (0.504 − 1.71i)14-s + (−0.959 + 0.137i)15-s + (0.601 + 0.694i)16-s + (0.0352 − 0.0161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.426629 + 0.301327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.426629 + 0.301327i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (-1.03e4 - 6.35e3i)T \) |
| good | 2 | \( 1 + (6.47 - 4.15i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (59.1 + 201. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (482. - 418. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (86.0 - 133. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-1.17e3 + 1.35e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-173. + 79.1i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-1.07e3 - 492. i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-1.31e4 - 2.87e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (146. + 1.01e3i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (1.30e4 - 4.44e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (1.05e5 - 3.09e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-1.23e5 - 1.77e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 + 4.50e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-8.50e4 + 7.36e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-2.53e5 + 2.92e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (1.93e4 - 2.78e3i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-3.44e4 - 5.36e4i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (3.33e4 - 2.14e4i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (2.00e5 - 4.39e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-6.45e4 - 5.59e4i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (2.39e5 - 8.16e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-6.50e5 - 9.35e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-2.16e5 - 7.37e5i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.13582918931268430911046563263, −12.77655779850546671665682116061, −11.94274266765597743757736382954, −9.696138779270759792640019201973, −8.820475743478337403804484073243, −8.246779573985328912875268496701, −6.77109598197542478512829219835, −5.39296951011592451615915851707, −3.33716611702609612710579347787, −0.923367953162726312672993231881,
0.36551656883934488845819759757, 2.80403147525687227303004821385, 3.93029078686350676771572575240, 6.29863120200614964707110093924, 7.38074086842392961152444858253, 9.062603426354992585341850090835, 10.26922688975317180627971729286, 10.54946788005548950901543820436, 11.60804166498348568265448798293, 13.55656715282053047259360640806
