| L(s) = 1 | + (−3.38 − 2.69i)2-s + (4.57 + 30.3i)3-s + (−10.0 − 44.1i)4-s + (1.30 + 1.90i)5-s + (66.3 − 114. i)6-s + (−187. + 108. i)7-s + (−205. + 426. i)8-s + (−202. + 62.3i)9-s + (0.747 − 9.96i)10-s + (409. − 1.79e3i)11-s + (1.29e3 − 507. i)12-s + (−136. − 1.81e3i)13-s + (925. + 139. i)14-s + (−51.9 + 48.1i)15-s + (−765. + 368. i)16-s + (−6.61e3 − 4.50e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.423 − 0.337i)2-s + (0.169 + 1.12i)3-s + (−0.157 − 0.689i)4-s + (0.0104 + 0.0152i)5-s + (0.307 − 0.532i)6-s + (−0.545 + 0.315i)7-s + (−0.400 + 0.832i)8-s + (−0.277 + 0.0855i)9-s + (0.000747 − 0.00996i)10-s + (0.308 − 1.34i)11-s + (0.747 − 0.293i)12-s + (−0.0619 − 0.827i)13-s + (0.337 + 0.0508i)14-s + (−0.0153 + 0.0142i)15-s + (−0.186 + 0.0899i)16-s + (−1.34 − 0.917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.404698 - 0.622595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.404698 - 0.622595i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.93e4 + 4.65e3i)T \) |
| good | 2 | \( 1 + (3.38 + 2.69i)T + (14.2 + 62.3i)T^{2} \) |
| 3 | \( 1 + (-4.57 - 30.3i)T + (-696. + 214. i)T^{2} \) |
| 5 | \( 1 + (-1.30 - 1.90i)T + (-5.70e3 + 1.45e4i)T^{2} \) |
| 7 | \( 1 + (187. - 108. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-409. + 1.79e3i)T + (-1.59e6 - 7.68e5i)T^{2} \) |
| 13 | \( 1 + (136. + 1.81e3i)T + (-4.77e6 + 7.19e5i)T^{2} \) |
| 17 | \( 1 + (6.61e3 + 4.50e3i)T + (8.81e6 + 2.24e7i)T^{2} \) |
| 19 | \( 1 + (-1.50e3 + 4.87e3i)T + (-3.88e7 - 2.65e7i)T^{2} \) |
| 23 | \( 1 + (5.84e3 + 5.42e3i)T + (1.10e7 + 1.47e8i)T^{2} \) |
| 29 | \( 1 + (1.46e3 - 9.73e3i)T + (-5.68e8 - 1.75e8i)T^{2} \) |
| 31 | \( 1 + (8.52e3 + 2.17e4i)T + (-6.50e8 + 6.03e8i)T^{2} \) |
| 37 | \( 1 + (-1.85e4 - 1.06e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (3.03e4 - 3.79e4i)T + (-1.05e9 - 4.63e9i)T^{2} \) |
| 47 | \( 1 + (-2.89e4 - 1.26e5i)T + (-9.71e9 + 4.67e9i)T^{2} \) |
| 53 | \( 1 + (-1.02e3 + 1.36e4i)T + (-2.19e10 - 3.30e9i)T^{2} \) |
| 59 | \( 1 + (-2.77e5 + 1.33e5i)T + (2.62e10 - 3.29e10i)T^{2} \) |
| 61 | \( 1 + (1.40e5 + 5.51e4i)T + (3.77e10 + 3.50e10i)T^{2} \) |
| 67 | \( 1 + (8.41e4 + 2.59e4i)T + (7.47e10 + 5.09e10i)T^{2} \) |
| 71 | \( 1 + (1.04e5 + 1.12e5i)T + (-9.57e9 + 1.27e11i)T^{2} \) |
| 73 | \( 1 + (2.79e5 - 2.09e4i)T + (1.49e11 - 2.25e10i)T^{2} \) |
| 79 | \( 1 + (4.63e5 + 8.03e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-8.30e5 + 1.25e5i)T + (3.12e11 - 9.63e10i)T^{2} \) |
| 89 | \( 1 + (7.69e4 + 5.10e5i)T + (-4.74e11 + 1.46e11i)T^{2} \) |
| 97 | \( 1 + (1.73e5 - 7.59e5i)T + (-7.50e11 - 3.61e11i)T^{2} \) |
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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
−14.52888588898500907988956067987, −13.30941150859518218348980526645, −11.42722298657437650502063753247, −10.51639381753195845719185722486, −9.462136579148833676032432685049, −8.667374257190525233347944535135, −6.22472935602026126580167374521, −4.76755134502830628734584592645, −2.92401539286441221307942481331, −0.38827314051644908944388193821,
1.84226902161461177840695418348, 4.02004499920606462428840830751, 6.69796021520132941273280365873, 7.26946839978202960799523328564, 8.619298452053721809668789526664, 9.890004984633952250168096350442, 11.90675422726769386846416989907, 12.80216676779773214661141411108, 13.55459495885508372799138741993, 15.14172624780086397226912802451
