| L(s) = 1 | + (−1.15 − 0.816i)2-s + (−0.857 − 0.514i)3-s + (0.665 + 1.88i)4-s + (−3.82 + 1.36i)5-s + (0.570 + 1.29i)6-s + (−4.41 + 1.34i)7-s + (0.771 − 2.72i)8-s + (0.471 + 0.881i)9-s + (5.52 + 1.54i)10-s + (1.56 − 2.10i)11-s + (0.398 − 1.95i)12-s + (−4.03 + 1.90i)13-s + (6.19 + 2.06i)14-s + (3.98 + 0.791i)15-s + (−3.11 + 2.51i)16-s + (−5.01 + 0.998i)17-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)2-s + (−0.495 − 0.296i)3-s + (0.332 + 0.942i)4-s + (−1.70 + 0.611i)5-s + (0.232 + 0.528i)6-s + (−1.66 + 0.506i)7-s + (0.272 − 0.962i)8-s + (0.157 + 0.293i)9-s + (1.74 + 0.487i)10-s + (0.471 − 0.635i)11-s + (0.115 − 0.565i)12-s + (−1.11 + 0.529i)13-s + (1.65 + 0.550i)14-s + (1.02 + 0.204i)15-s + (−0.778 + 0.627i)16-s + (−1.21 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134352 - 0.0938456i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134352 - 0.0938456i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 + 0.816i)T \) |
| 3 | \( 1 + (0.857 + 0.514i)T \) |
| good | 5 | \( 1 + (3.82 - 1.36i)T + (3.86 - 3.17i)T^{2} \) |
| 7 | \( 1 + (4.41 - 1.34i)T + (5.82 - 3.88i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 2.10i)T + (-3.19 - 10.5i)T^{2} \) |
| 13 | \( 1 + (4.03 - 1.90i)T + (8.24 - 10.0i)T^{2} \) |
| 17 | \( 1 + (5.01 - 0.998i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-0.300 - 6.11i)T + (-18.9 + 1.86i)T^{2} \) |
| 23 | \( 1 + (-0.0676 - 0.686i)T + (-22.5 + 4.48i)T^{2} \) |
| 29 | \( 1 + (-1.04 + 7.07i)T + (-27.7 - 8.41i)T^{2} \) |
| 31 | \( 1 + (-0.0538 - 0.0223i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (1.37 + 1.51i)T + (-3.62 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-3.65 - 2.99i)T + (7.99 + 40.2i)T^{2} \) |
| 43 | \( 1 + (2.44 + 4.08i)T + (-20.2 + 37.9i)T^{2} \) |
| 47 | \( 1 + (-4.11 - 2.74i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (1.30 + 8.81i)T + (-50.7 + 15.3i)T^{2} \) |
| 59 | \( 1 + (5.18 + 2.45i)T + (37.4 + 45.6i)T^{2} \) |
| 61 | \( 1 + (13.9 - 3.49i)T + (53.7 - 28.7i)T^{2} \) |
| 67 | \( 1 + (-1.40 - 5.61i)T + (-59.0 + 31.5i)T^{2} \) |
| 71 | \( 1 + (-2.72 - 1.45i)T + (39.4 + 59.0i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 3.29i)T + (60.6 + 40.5i)T^{2} \) |
| 79 | \( 1 + (-6.13 - 9.18i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (3.31 - 3.65i)T + (-8.13 - 82.6i)T^{2} \) |
| 89 | \( 1 + (-0.530 + 5.38i)T + (-87.2 - 17.3i)T^{2} \) |
| 97 | \( 1 + (-4.41 + 10.6i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−10.19537360155366158141133399451, −9.408011403696363816380073532357, −8.454241482804417894453600788529, −7.60001651012315373575561579318, −6.80080309151462022169633105182, −6.21531390577561407428704123203, −4.21702765410095262979537802594, −3.49812393178860647660514861753, −2.46613588529788194544745012787, −0.23826267467306305859468891476,
0.55102733198326317333007302274, 3.01908415517960998012680125790, 4.36189836165085588600983546604, 4.98626914205349721628359110624, 6.55544161424678907151152757428, 7.04562332002815869114542384712, 7.71838206988369390605500421222, 9.142410476087565047480885929086, 9.240290557693804098327336444140, 10.48139139685879047726942216445
