| L(s) = 1 | + (−10.0 + 3.09i)2-s + (9.56 + 24.3i)3-s + (64.7 − 44.1i)4-s + (−25.7 − 3.88i)5-s + (−171. − 214. i)6-s + (24.4 − 127. i)7-s + (−303. + 381. i)8-s + (−323. + 300. i)9-s + (270. − 40.8i)10-s + (−479. − 444. i)11-s + (1.69e3 + 1.15e3i)12-s + (195. − 855. i)13-s + (148. + 1.35e3i)14-s + (−151. − 665. i)15-s + (954. − 2.43e3i)16-s + (59.6 − 795. i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.547i)2-s + (0.613 + 1.56i)3-s + (2.02 − 1.37i)4-s + (−0.461 − 0.0695i)5-s + (−1.94 − 2.43i)6-s + (0.188 − 0.981i)7-s + (−1.67 + 2.10i)8-s + (−1.33 + 1.23i)9-s + (0.856 − 0.129i)10-s + (−1.19 − 1.10i)11-s + (3.39 + 2.31i)12-s + (0.320 − 1.40i)13-s + (0.202 + 1.84i)14-s + (−0.174 − 0.763i)15-s + (0.932 − 2.37i)16-s + (0.0500 − 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.375858 - 0.141907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375858 - 0.141907i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-24.4 + 127. i)T \) |
| good | 2 | \( 1 + (10.0 - 3.09i)T + (26.4 - 18.0i)T^{2} \) |
| 3 | \( 1 + (-9.56 - 24.3i)T + (-178. + 165. i)T^{2} \) |
| 5 | \( 1 + (25.7 + 3.88i)T + (2.98e3 + 921. i)T^{2} \) |
| 11 | \( 1 + (479. + 444. i)T + (1.20e4 + 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-195. + 855. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-59.6 + 795. i)T + (-1.40e6 - 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-702. - 1.21e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-68.6 - 916. i)T + (-6.36e6 + 9.59e5i)T^{2} \) |
| 29 | \( 1 + (675. - 325. i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-18.1 + 31.3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-132. - 90.4i)T + (2.53e7 + 6.45e7i)T^{2} \) |
| 41 | \( 1 + (-5.24e3 + 6.57e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (6.76e3 + 8.48e3i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.24e4 - 3.83e3i)T + (1.89e8 - 1.29e8i)T^{2} \) |
| 53 | \( 1 + (1.22e4 - 8.37e3i)T + (1.52e8 - 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-2.58e4 + 3.89e3i)T + (6.83e8 - 2.10e8i)T^{2} \) |
| 61 | \( 1 + (2.29e4 + 1.56e4i)T + (3.08e8 + 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-3.00e4 + 5.20e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.70e4 + 8.21e3i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.02e3 + 624. i)T + (1.71e9 + 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-3.91e4 - 6.77e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.72e4 + 7.55e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-3.37e4 + 3.13e4i)T + (4.17e8 - 5.56e9i)T^{2} \) |
| 97 | \( 1 + 9.73e4T + 8.58e9T^{2} \) |
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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
−15.17999885518821810982097300666, −13.85394167494004377441141809112, −11.09928538019523300811651963753, −10.48700568846243969189668273114, −9.666681012579656374484098612421, −8.272326236461238606864526072863, −7.77961443027511085894103209013, −5.47128578797054223007444508553, −3.28144114889075762354747585598, −0.33005915453094066532964352491,
1.66399818998068330966598013923, 2.57859113725447863802435519071, 6.70209698434516896270569473196, 7.70457689410854960316950852240, 8.477156765634679954102608857971, 9.572030135675138606931131843637, 11.33562143929035626912843812881, 12.11781537729962330689906616827, 13.09322546957983288006731407001, 14.94310704366504725867209332754
