| L(s) = 1 | + (−2 + 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (37.7 − 65.3i)5-s − 36·6-s + (99.4 − 83.1i)7-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (150. + 261. i)10-s + (74.7 + 129. i)11-s + (72 − 124. i)12-s + 349.·13-s + (88.9 + 510. i)14-s + 679.·15-s + (−128 + 221. i)16-s + (574. + 995. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.675 − 1.16i)5-s − 0.408·6-s + (0.767 − 0.641i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.477 + 0.826i)10-s + (0.186 + 0.322i)11-s + (0.144 − 0.249i)12-s + 0.573·13-s + (0.121 + 0.696i)14-s + 0.779·15-s + (−0.125 + 0.216i)16-s + (0.482 + 0.835i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.71801 + 0.203520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71801 + 0.203520i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (-99.4 + 83.1i)T \) |
| good | 5 | \( 1 + (-37.7 + 65.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-74.7 - 129. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-574. - 995. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.39e3 + 2.42e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (906. - 1.57e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 759.T + 2.05e7T^{2} \) |
| 31 | \( 1 + (4.51e3 + 7.82e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.89e3 - 6.75e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.22e4 - 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (6.79e3 + 1.17e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.31e4 - 2.28e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.76e4 - 3.05e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.71e4 + 4.70e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.22e4 - 3.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.08e4 - 5.33e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 8.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (4.92e4 - 8.53e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.23e4T + 8.58e9T^{2} \) |
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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
−15.17631296592097376243115542764, −13.97107247357890695106260512512, −13.08246783103391748691891156258, −11.21851718098413744130308417001, −9.775010148569778627993517380606, −8.862852256654795749360621217304, −7.62893463278246096588817209785, −5.63086540738513524545537447771, −4.39841466560044460160635738177, −1.29799480292732588237456909053,
1.72159539290407292860288789661, 3.17332306283888373554353941637, 5.81109102927982809547068560307, 7.43769984278682279682959021933, 8.794982447906053593213236342280, 10.17810844914614595438253428150, 11.32016955393892164608015431388, 12.40875774893812824791127309990, 14.07119603961883638956409695872, 14.40762082674568385007892034832
