| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−1.53 − 1.28i)4-s + (2.99 + 0.528i)5-s + (6.30 − 5.28i)7-s + (2.44 − 1.41i)8-s + (−2.15 + 3.72i)10-s + (−6.45 + 1.13i)11-s + (17.4 − 6.33i)13-s + (3.98 + 10.9i)14-s + (0.694 + 3.93i)16-s + (11.4 + 6.59i)17-s + (17.4 + 30.1i)19-s + (−3.91 − 4.66i)20-s + (1.60 − 9.12i)22-s + (−2.12 + 2.53i)23-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.383 − 0.321i)4-s + (0.599 + 0.105i)5-s + (0.900 − 0.755i)7-s + (0.306 − 0.176i)8-s + (−0.215 + 0.372i)10-s + (−0.586 + 0.103i)11-s + (1.33 − 0.487i)13-s + (0.284 + 0.781i)14-s + (0.0434 + 0.246i)16-s + (0.672 + 0.388i)17-s + (0.916 + 1.58i)19-s + (−0.195 − 0.233i)20-s + (0.0731 − 0.414i)22-s + (−0.0926 + 0.110i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46476 + 0.422030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46476 + 0.422030i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.99 - 0.528i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.30 + 5.28i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (6.45 - 1.13i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 6.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-11.4 - 6.59i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-17.4 - 30.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (2.12 - 2.53i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-9.72 + 26.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (10.6 + 8.92i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (3.33 - 5.77i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-19.4 - 53.5i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (5.68 + 32.2i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (34.5 + 41.1i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 98.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (101. + 17.8i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (5.50 - 4.61i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (28.2 - 10.2i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (8.46 + 4.88i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-64.7 - 112. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (96.4 + 35.1i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (37.0 - 101. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-20.1 + 11.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.89 + 50.4i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.05005802820932472025714067318, −11.58161114642302420381555086999, −10.44264663350503941874950486492, −9.818201310920910681525237675013, −8.192266978773368770234053101312, −7.77922349726054172135827136743, −6.21157230824474347467630144856, −5.34144055390822668307386710116, −3.78764842464822441340669536893, −1.42241190344081692903603925101,
1.49824199641295377215559155382, 2.98074942138490422814210658531, 4.80135909393716665264446711828, 5.83256085732788406080037061526, 7.52849026455990617040679608512, 8.744862483743857485879735790977, 9.368968135778694707177473430031, 10.74616469202003871479559408655, 11.40370220427033290075466175945, 12.39909881573363976489063363424
