| L(s) = 1 | + (1.39 + 0.245i)2-s + (1.87 + 0.684i)4-s + (−5.54 − 6.61i)5-s + (7.83 − 2.85i)7-s + (2.44 + 1.41i)8-s + (−6.10 − 10.5i)10-s + (10.8 − 12.8i)11-s + (0.524 + 2.97i)13-s + (11.6 − 2.04i)14-s + (3.06 + 2.57i)16-s + (−8.43 + 4.87i)17-s + (3.84 − 6.66i)19-s + (−5.90 − 16.2i)20-s + (18.2 − 15.3i)22-s + (−10.1 + 27.8i)23-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.469 + 0.171i)4-s + (−1.10 − 1.32i)5-s + (1.11 − 0.407i)7-s + (0.306 + 0.176i)8-s + (−0.610 − 1.05i)10-s + (0.983 − 1.17i)11-s + (0.0403 + 0.229i)13-s + (0.829 − 0.146i)14-s + (0.191 + 0.160i)16-s + (−0.496 + 0.286i)17-s + (0.202 − 0.350i)19-s + (−0.295 − 0.811i)20-s + (0.828 − 0.695i)22-s + (−0.441 + 1.21i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85702 - 0.853537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85702 - 0.853537i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.245i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (5.54 + 6.61i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-7.83 + 2.85i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (-10.8 + 12.8i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-0.524 - 2.97i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (8.43 - 4.87i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.84 + 6.66i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (10.1 - 27.8i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-10.5 - 1.86i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (10.8 + 3.93i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (11.5 + 20.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-16.7 + 2.94i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-18.0 - 15.1i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-5.67 - 15.6i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 - 75.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-38.6 - 46.1i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-32.1 + 11.6i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-6.53 - 37.0i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (44.0 - 25.4i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-49.2 + 85.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.0 - 79.9i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-15.8 - 2.80i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (14.0 + 8.09i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (101. + 85.2i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−12.42070745162974735371138191298, −11.54104482913747239850953448763, −11.10609466564462748445254080047, −9.082039239986452613205809356972, −8.279973712198834329021267383349, −7.35285946255479719904214690344, −5.71969864558991074202872013535, −4.53421046246794290880487553843, −3.79304443628825642078591797344, −1.20801473193519421155185790034,
2.24751423620206645855222482862, 3.78339640830909753305127774043, 4.78578237469963540949058628909, 6.50017117691484432626586534198, 7.32507624539231951415747492499, 8.394903901559762646463046449419, 10.08638358645595093698983983415, 11.11724042417377198433757827088, 11.75057648642331566481182461196, 12.46113039757558868473821095082
