| L(s) = 1 | + (−4.41 − 19.3i)2-s + (54.3 + 50.4i)3-s + (−238. + 114. i)4-s + (−106. + 16.0i)5-s + (734. − 1.27e3i)6-s + (−516. − 895. i)7-s + (1.69e3 + 2.12e3i)8-s + (246. + 3.29e3i)9-s + (777. + 1.98e3i)10-s + (−2.47e3 − 1.18e3i)11-s + (−1.87e4 − 5.78e3i)12-s + (−5.58e3 + 1.42e4i)13-s + (−1.50e4 + 1.39e4i)14-s + (−6.57e3 − 4.48e3i)15-s + (1.24e4 − 1.55e4i)16-s + (1.33e3 + 201. i)17-s + ⋯ |
| L(s) = 1 | + (−0.389 − 1.70i)2-s + (1.16 + 1.07i)3-s + (−1.86 + 0.898i)4-s + (−0.379 + 0.0572i)5-s + (1.38 − 2.40i)6-s + (−0.569 − 0.986i)7-s + (1.16 + 1.46i)8-s + (0.112 + 1.50i)9-s + (0.245 + 0.626i)10-s + (−0.559 − 0.269i)11-s + (−3.13 − 0.966i)12-s + (−0.705 + 1.79i)13-s + (−1.46 + 1.35i)14-s + (−0.502 − 0.342i)15-s + (0.757 − 0.949i)16-s + (0.0659 + 0.00993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.349668 + 0.302430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.349668 + 0.302430i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.21e5 - 1.34e4i)T \) |
| good | 2 | \( 1 + (4.41 + 19.3i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-54.3 - 50.4i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (106. - 16.0i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (516. + 895. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.47e3 + 1.18e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (5.58e3 - 1.42e4i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-1.33e3 - 201. i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (2.27e3 - 3.03e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (4.21e4 - 2.87e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-8.45e4 + 7.84e4i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (2.34e5 + 7.24e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-3.46e4 + 5.99e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.48e5 + 6.51e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (8.81e5 - 4.24e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-6.41e5 - 1.63e6i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (-1.46e5 + 1.84e5i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (3.18e5 - 9.82e4i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-2.14e5 + 2.86e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (-1.53e6 - 1.04e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (-1.18e6 + 3.03e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (-1.77e6 - 3.06e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.58e6 + 2.39e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (3.69e6 + 3.42e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (5.47e6 + 2.63e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−14.27044967411768005540410209209, −13.57338983189650332606983214209, −12.08430829796143335439369808534, −10.79256665637014081234238276404, −9.862529636963315173044441220190, −9.203118285838386654689452650338, −7.77844590328505774629813423172, −4.19498287735515351653261920073, −3.63160649135998941852015979778, −2.12424742653803030716652769969,
0.18170243361118872682551690986, 2.74811221683440564873789377222, 5.39574675955597727035180706742, 6.82528396410811364938453621429, 7.87846465192589767526777289397, 8.501563621042789421014200152809, 9.746397859742720730791128270656, 12.51359109621375790290485688435, 13.25145417313841739157650474962, 14.61167024418255235084167005862
