L(s) = 1 | + (0.766 − 0.642i)2-s + (1.57 + 0.726i)3-s + (0.173 − 0.984i)4-s + (−1.96 + 0.346i)5-s + (1.67 − 0.453i)6-s + (0.910 − 1.57i)7-s + (−0.500 − 0.866i)8-s + (1.94 + 2.28i)9-s + (−1.28 + 1.52i)10-s + (−4.10 + 2.37i)11-s + (0.988 − 1.42i)12-s + (0.151 + 0.415i)13-s + (−0.316 − 1.79i)14-s + (−3.34 − 0.883i)15-s + (−0.939 − 0.342i)16-s + (−1.07 − 1.28i)17-s + ⋯ |
L(s) = 1 | + (0.541 − 0.454i)2-s + (0.907 + 0.419i)3-s + (0.0868 − 0.492i)4-s + (−0.879 + 0.155i)5-s + (0.682 − 0.185i)6-s + (0.344 − 0.596i)7-s + (−0.176 − 0.306i)8-s + (0.647 + 0.761i)9-s + (−0.405 + 0.483i)10-s + (−1.23 + 0.715i)11-s + (0.285 − 0.410i)12-s + (0.0419 + 0.115i)13-s + (−0.0845 − 0.479i)14-s + (−0.863 − 0.228i)15-s + (−0.234 − 0.0855i)16-s + (−0.260 − 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49621 - 0.258165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49621 - 0.258165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-1.57 - 0.726i)T \) |
| 19 | \( 1 + (3.58 + 2.48i)T \) |
good | 5 | \( 1 + (1.96 - 0.346i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.910 + 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.10 - 2.37i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.151 - 0.415i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.07 + 1.28i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-5.93 - 1.04i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.91 + 4.12i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.88 - 2.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 + (-3.75 - 1.36i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.15 - 12.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.92 + 8.25i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.424 + 2.40i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.87 + 3.24i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.80 + 10.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.27 + 6.28i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.897 - 5.08i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.5 + 4.94i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.23 - 8.88i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.523 + 0.302i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 1.48i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.64 - 1.96i)T + (-16.8 + 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.44745848684962061472901996066, −12.78076046405316412276267741650, −11.28225463726360339545077380715, −10.58754978478886094180862602943, −9.413073016544847677072250199827, −8.016662358610621566038695409093, −7.16765443588319320242731379722, −4.91869151987006718148784347466, −4.00486293714043435025475025645, −2.55037531086282043472881374126,
2.71185842033480500845573931040, 4.12726707709151827256796262475, 5.68468573091249379506794748470, 7.23431799778637783113846043850, 8.193993943195314546999435137914, 8.789550721009369979572453258924, 10.65368682105266176963011858343, 11.92923526652684159073314435834, 12.83825116609298959338208667649, 13.58974974959484535616169150072