| L(s) = 1 | + (5.65 + 9.79i)2-s + (−63.9 + 110. i)4-s + (−32.9 + 18.9i)5-s + (1.39e3 + 1.95e3i)7-s − 1.44e3·8-s + (−372. − 214. i)10-s + (7.79e3 − 1.35e4i)11-s + 2.41e4i·13-s + (−1.11e4 + 2.47e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (1.16e5 + 6.73e4i)17-s + (1.68e4 − 9.70e3i)19-s − 4.86e3i·20-s + 1.76e5·22-s + (−1.93e5 − 3.34e5i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0526 + 0.0303i)5-s + (0.583 + 0.812i)7-s − 0.353·8-s + (−0.0372 − 0.0214i)10-s + (0.532 − 0.922i)11-s + 0.844i·13-s + (−0.291 + 0.644i)14-s + (−0.125 − 0.216i)16-s + (1.39 + 0.806i)17-s + (0.129 − 0.0744i)19-s − 0.0303i·20-s + 0.753·22-s + (−0.690 − 1.19i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.400417896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400417896\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 - 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.39e3 - 1.95e3i)T \) |
| good | 5 | \( 1 + (32.9 - 18.9i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-7.79e3 + 1.35e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.41e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.16e5 - 6.73e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.68e4 + 9.70e3i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.93e5 + 3.34e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.53e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-5.11e5 - 2.95e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-2.85e5 - 4.94e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 4.86e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.56e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (1.97e6 - 1.13e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-1.96e5 + 3.41e5i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.17e7 - 6.78e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.72e6 + 3.88e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.77e7 - 3.07e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.53e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (3.46e7 + 2.00e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.81e7 - 3.13e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 2.35e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.04e7 - 1.75e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 4.41e7iT - 7.83e15T^{2} \) |
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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
−12.08229352096607597100977691229, −11.53345617094443142510498147317, −10.00802955456702285262912624678, −8.685610438851322883219542053618, −8.059125363068878465269956702215, −6.54214356727275843026135181309, −5.68034832267934171866624400845, −4.42597496094827536998273092760, −3.08130651787987469917427825487, −1.39047384203577809122285226054,
0.59533734169480750772385766093, 1.74963964696900749937389130797, 3.33638887856558433212121242097, 4.46256441779387250576098679393, 5.57611630880275745495199936001, 7.16850441059453659735683276484, 8.150188167830339631609291636498, 9.791109124218377232602020628340, 10.28690460222024546709806049034, 11.67302541616572656852522878543
