| L(s) = 1 | + (−54.0 − 93.6i)2-s + (8.48e3 + 4.89e3i)3-s + (2.69e4 − 4.66e4i)4-s + (2.14e5 − 1.23e5i)5-s − 1.06e6i·6-s + (5.16e5 + 5.74e6i)7-s − 1.29e7·8-s + (2.64e7 + 4.58e7i)9-s + (−2.31e7 − 1.33e7i)10-s + (2.02e8 − 3.49e8i)11-s + (4.56e8 − 2.63e8i)12-s + 7.03e8i·13-s + (5.09e8 − 3.58e8i)14-s + 2.42e9·15-s + (−1.06e9 − 1.84e9i)16-s + (5.78e9 + 3.33e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.211 − 0.365i)2-s + (1.29 + 0.746i)3-s + (0.410 − 0.711i)4-s + (0.547 − 0.316i)5-s − 0.631i·6-s + (0.0896 + 0.995i)7-s − 0.769·8-s + (0.615 + 1.06i)9-s + (−0.231 − 0.133i)10-s + (0.942 − 1.63i)11-s + (1.06 − 0.613i)12-s + 0.862i·13-s + (0.345 − 0.243i)14-s + 0.944·15-s + (−0.248 − 0.429i)16-s + (0.828 + 0.478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(2.85001 - 0.441087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.85001 - 0.441087i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.16e5 - 5.74e6i)T \) |
| good | 2 | \( 1 + (54.0 + 93.6i)T + (-3.27e4 + 5.67e4i)T^{2} \) |
| 3 | \( 1 + (-8.48e3 - 4.89e3i)T + (2.15e7 + 3.72e7i)T^{2} \) |
| 5 | \( 1 + (-2.14e5 + 1.23e5i)T + (7.62e10 - 1.32e11i)T^{2} \) |
| 11 | \( 1 + (-2.02e8 + 3.49e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 13 | \( 1 - 7.03e8iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (-5.78e9 - 3.33e9i)T + (2.43e19 + 4.21e19i)T^{2} \) |
| 19 | \( 1 + (-1.30e10 + 7.54e9i)T + (1.44e20 - 2.49e20i)T^{2} \) |
| 23 | \( 1 + (1.85e9 + 3.21e9i)T + (-3.06e21 + 5.31e21i)T^{2} \) |
| 29 | \( 1 + 6.34e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (-7.23e10 - 4.17e10i)T + (3.63e23 + 6.29e23i)T^{2} \) |
| 37 | \( 1 + (2.10e12 + 3.65e12i)T + (-6.16e24 + 1.06e25i)T^{2} \) |
| 41 | \( 1 - 8.05e12iT - 6.37e25T^{2} \) |
| 43 | \( 1 + 1.17e13T + 1.36e26T^{2} \) |
| 47 | \( 1 + (4.21e12 - 2.43e12i)T + (2.83e26 - 4.91e26i)T^{2} \) |
| 53 | \( 1 + (-1.17e11 + 2.02e11i)T + (-1.93e27 - 3.35e27i)T^{2} \) |
| 59 | \( 1 + (7.78e12 + 4.49e12i)T + (1.07e28 + 1.86e28i)T^{2} \) |
| 61 | \( 1 + (2.29e14 - 1.32e14i)T + (1.83e28 - 3.18e28i)T^{2} \) |
| 67 | \( 1 + (1.76e14 - 3.05e14i)T + (-8.24e28 - 1.42e29i)T^{2} \) |
| 71 | \( 1 - 2.43e13T + 4.16e29T^{2} \) |
| 73 | \( 1 + (-8.48e14 - 4.89e14i)T + (3.25e29 + 5.63e29i)T^{2} \) |
| 79 | \( 1 + (1.08e15 + 1.87e15i)T + (-1.15e30 + 1.99e30i)T^{2} \) |
| 83 | \( 1 - 3.59e14iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (1.33e15 - 7.71e14i)T + (7.74e30 - 1.34e31i)T^{2} \) |
| 97 | \( 1 + 1.09e16iT - 6.14e31T^{2} \) |
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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
−18.79336550176393383122968208147, −16.33923510683726084400900436613, −14.92502730290856272796888137529, −13.93563382154922214875644452855, −11.47558852699524555774833029400, −9.548568851394656689418072842462, −8.825640811533336648599662589311, −5.76998154964579201267876224357, −3.23401455634325516103887493040, −1.64310745755055165630467904564,
1.76554663625915233301719149501, 3.39419938961156604965161512412, 6.98531220848897992318780661231, 7.83000886875655450345275943813, 9.707037009866405621882921299375, 12.34111768433267883685306372070, 13.79203707858730622105750931051, 14.99132845447831931943725412990, 17.04936178355061598115593435354, 18.18149980533936273498208937738
