| L(s) = 1 | + (0.644 − 1.25i)2-s + (−0.526 − 0.253i)3-s + (−1.16 − 1.62i)4-s + (−2.60 + 0.594i)5-s + (−0.658 + 0.499i)6-s + (−1.20 − 0.578i)7-s + (−2.79 + 0.424i)8-s + (−1.65 − 2.07i)9-s + (−0.930 + 3.65i)10-s + (0.405 − 0.508i)11-s + (0.203 + 1.15i)12-s + (−0.285 − 0.227i)13-s + (−1.50 + 1.13i)14-s + (1.52 + 0.347i)15-s + (−1.26 + 3.79i)16-s − 4.19i·17-s + ⋯ |
| L(s) = 1 | + (0.455 − 0.890i)2-s + (−0.303 − 0.146i)3-s + (−0.584 − 0.811i)4-s + (−1.16 + 0.265i)5-s + (−0.268 + 0.203i)6-s + (−0.454 − 0.218i)7-s + (−0.988 + 0.150i)8-s + (−0.552 − 0.692i)9-s + (−0.294 + 1.15i)10-s + (0.122 − 0.153i)11-s + (0.0588 + 0.332i)12-s + (−0.0792 − 0.0632i)13-s + (−0.401 + 0.304i)14-s + (0.392 + 0.0896i)15-s + (−0.317 + 0.948i)16-s − 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0337949 + 0.652823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0337949 + 0.652823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.644 + 1.25i)T \) |
| 29 | \( 1 + (2.63 + 4.69i)T \) |
| good | 3 | \( 1 + (0.526 + 0.253i)T + (1.87 + 2.34i)T^{2} \) |
| 5 | \( 1 + (2.60 - 0.594i)T + (4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (1.20 + 0.578i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.405 + 0.508i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.285 + 0.227i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + 4.19iT - 17T^{2} \) |
| 19 | \( 1 + (-4.67 + 2.24i)T + (11.8 - 14.8i)T^{2} \) |
| 23 | \( 1 + (0.460 - 2.01i)T + (-20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-4.16 + 0.950i)T + (27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (-0.0392 - 0.0492i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + 8.41iT - 41T^{2} \) |
| 43 | \( 1 + (0.637 - 2.79i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.218 + 0.174i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (4.44 - 1.01i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 3.70iT - 59T^{2} \) |
| 61 | \( 1 + (6.12 + 2.95i)T + (38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 8.35i)T + (14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (10.2 - 12.8i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-13.1 - 2.99i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + (-10.9 + 8.73i)T + (17.5 - 77.0i)T^{2} \) |
| 83 | \( 1 + (4.58 + 9.51i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (3.59 - 0.821i)T + (80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (0.786 + 1.63i)T + (-60.4 + 75.8i)T^{2} \) |
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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
−11.62167055850021902398691947387, −11.23501017382460628253899262877, −9.882812559595095524712856912812, −9.044358653493481591489787839906, −7.63971318573857913896752730894, −6.47298301566738734342547279925, −5.21970613319241242812644427261, −3.85165065098054170282400576067, −2.98637451821502679636053666636, −0.47840516122923223543330127687,
3.25771618053931521222364298560, 4.40669226068663494980480576370, 5.45931916391197285784550900579, 6.56680833013885212174675362565, 7.82179240168360260469786692697, 8.344101998623774964753155763954, 9.594124371860331049394600479082, 10.99916580348875533887502319970, 12.00144858283812351816333757715, 12.58048588426887712160502781590
