| L(s) = 1 | + (−0.839 − 1.13i)2-s + (1.61 + 0.615i)3-s + (−0.589 + 1.91i)4-s + (1.17 + 0.102i)5-s + (−0.659 − 2.35i)6-s + (0.105 + 0.288i)7-s + (2.66 − 0.934i)8-s + (2.24 + 1.99i)9-s + (−0.870 − 1.42i)10-s + (0.523 + 5.98i)11-s + (−2.13 + 2.73i)12-s + (−2.00 − 1.40i)13-s + (0.240 − 0.362i)14-s + (1.84 + 0.890i)15-s + (−3.30 − 2.25i)16-s + (0.322 + 0.558i)17-s + ⋯ |
| L(s) = 1 | + (−0.593 − 0.804i)2-s + (0.934 + 0.355i)3-s + (−0.294 + 0.955i)4-s + (0.525 + 0.0460i)5-s + (−0.269 − 0.963i)6-s + (0.0397 + 0.109i)7-s + (0.943 − 0.330i)8-s + (0.747 + 0.664i)9-s + (−0.275 − 0.450i)10-s + (0.157 + 1.80i)11-s + (−0.614 + 0.788i)12-s + (−0.555 − 0.388i)13-s + (0.0642 − 0.0968i)14-s + (0.475 + 0.229i)15-s + (−0.826 − 0.563i)16-s + (0.0782 + 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48945 + 0.0971426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48945 + 0.0971426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.839 + 1.13i)T \) |
| 3 | \( 1 + (-1.61 - 0.615i)T \) |
| good | 5 | \( 1 + (-1.17 - 0.102i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-0.105 - 0.288i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.523 - 5.98i)T + (-10.8 + 1.91i)T^{2} \) |
| 13 | \( 1 + (2.00 + 1.40i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.322 - 0.558i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.197 + 0.736i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.965 + 2.65i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.94 - 5.62i)T + (-9.91 + 27.2i)T^{2} \) |
| 31 | \( 1 + (-3.42 - 1.24i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.38 + 8.88i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.59 + 0.457i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.247 + 2.83i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.08 + 2.21i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (9.46 + 9.46i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.05 + 0.442i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (1.99 - 4.26i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (1.35 + 0.949i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-7.04 + 4.06i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.57 + 4.37i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.106 - 0.603i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.20 - 10.2i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (10.8 + 6.23i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.11 + 6.80i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.79580010081240662037427360430, −10.06952106185610142833170158596, −9.548739342221654759913754245234, −8.727751636010539190849144054346, −7.69767028792524227398682148629, −6.94886935361045753223054165437, −4.98426196211050217261321529721, −4.07763306916303095404991846066, −2.68690365709747820500586112620, −1.85944518696855589961981627400,
1.18951968969476359326867149886, 2.78138030113645777479053001468, 4.33482405324029275139618934235, 5.82596082132581016136192487726, 6.49243312429016910602323230983, 7.70230259090500860952746806771, 8.284545175633670750047017674905, 9.248682362094427716598895647210, 9.778598012753350887015483149880, 10.89760517263349010385438731450
