| L(s) = 1 | + (−0.410 − 0.852i)2-s + (−16.8 + 1.26i)3-s + (9.41 − 11.8i)4-s + (6.39 + 20.7i)5-s + (7.98 + 13.8i)6-s + (31.1 + 17.9i)7-s + (−28.7 − 6.55i)8-s + (201. − 30.3i)9-s + (15.0 − 13.9i)10-s + (139. + 174. i)11-s + (−143. + 210. i)12-s + (−60.7 − 56.4i)13-s + (2.54 − 33.9i)14-s + (−133. − 340. i)15-s + (−47.5 − 208. i)16-s + (67.3 + 20.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.102 − 0.213i)2-s + (−1.86 + 0.140i)3-s + (0.588 − 0.738i)4-s + (0.255 + 0.828i)5-s + (0.221 + 0.383i)6-s + (0.636 + 0.367i)7-s + (−0.448 − 0.102i)8-s + (2.48 − 0.374i)9-s + (0.150 − 0.139i)10-s + (1.15 + 1.44i)11-s + (−0.996 + 1.46i)12-s + (−0.359 − 0.333i)13-s + (0.0129 − 0.173i)14-s + (−0.593 − 1.51i)15-s + (−0.185 − 0.814i)16-s + (0.232 + 0.0718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.954434 + 0.256328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.954434 + 0.256328i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-472. - 1.78e3i)T \) |
| good | 2 | \( 1 + (0.410 + 0.852i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (16.8 - 1.26i)T + (80.0 - 12.0i)T^{2} \) |
| 5 | \( 1 + (-6.39 - 20.7i)T + (-516. + 352. i)T^{2} \) |
| 7 | \( 1 + (-31.1 - 17.9i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-139. - 174. i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (60.7 + 56.4i)T + (2.13e3 + 2.84e4i)T^{2} \) |
| 17 | \( 1 + (-67.3 - 20.7i)T + (6.90e4 + 4.70e4i)T^{2} \) |
| 19 | \( 1 + (62.0 - 411. i)T + (-1.24e5 - 3.84e4i)T^{2} \) |
| 23 | \( 1 + (57.4 - 146. i)T + (-2.05e5 - 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-1.20e3 - 90.5i)T + (6.99e5 + 1.05e5i)T^{2} \) |
| 31 | \( 1 + (-571. - 389. i)T + (3.37e5 + 8.59e5i)T^{2} \) |
| 37 | \( 1 + (664. - 383. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.31e3 - 633. i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-1.22e3 + 1.54e3i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (-2.80e3 + 2.60e3i)T + (5.89e5 - 7.86e6i)T^{2} \) |
| 59 | \( 1 + (-269. - 1.18e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (1.13e3 + 1.66e3i)T + (-5.05e6 + 1.28e7i)T^{2} \) |
| 67 | \( 1 + (7.96e3 + 1.20e3i)T + (1.92e7 + 5.93e6i)T^{2} \) |
| 71 | \( 1 + (3.94e3 - 1.54e3i)T + (1.86e7 - 1.72e7i)T^{2} \) |
| 73 | \( 1 + (-876. + 944. i)T + (-2.12e6 - 2.83e7i)T^{2} \) |
| 79 | \( 1 + (-4.21e3 + 7.30e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-121. - 1.62e3i)T + (-4.69e7 + 7.07e6i)T^{2} \) |
| 89 | \( 1 + (4.17e3 - 312. i)T + (6.20e7 - 9.35e6i)T^{2} \) |
| 97 | \( 1 + (-577. - 724. i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.28059595382354072558144697116, −14.54622005739224714701204505712, −12.15121472832268333112071154996, −11.79807004560532194904199218179, −10.45219271340441393986506346961, −9.985437009913409787517882782868, −6.99401450466819765799592018123, −6.17159580006454174686629838039, −4.85178816890912048617625373686, −1.54557215560893486417476078469,
0.954144164416155490025254721793, 4.51851914767435032506758020658, 5.98516237470700873459823665086, 7.06244778226089085558783908598, 8.765679477781908858823382958614, 10.75177525772356319032981814577, 11.65339048522799485751866279772, 12.29360060922052833774823962733, 13.66467041354691335426070051577, 15.71593636824541203538504687301
