L(s) = 1 | + (−0.917 + 1.07i)2-s + (−0.843 − 1.51i)3-s + (−0.317 − 1.97i)4-s + (−3.45 − 1.84i)5-s + (2.40 + 0.478i)6-s + (−1.46 − 0.291i)7-s + (2.41 + 1.46i)8-s + (−1.57 + 2.55i)9-s + (5.15 − 2.02i)10-s + (0.819 − 0.672i)11-s + (−2.71 + 2.14i)12-s + (2.93 − 1.57i)13-s + (1.65 − 1.31i)14-s + (0.121 + 6.78i)15-s + (−3.79 + 1.25i)16-s + (−2.84 + 1.17i)17-s + ⋯ |
L(s) = 1 | + (−0.648 + 0.761i)2-s + (−0.487 − 0.873i)3-s + (−0.158 − 0.987i)4-s + (−1.54 − 0.825i)5-s + (0.980 + 0.195i)6-s + (−0.554 − 0.110i)7-s + (0.854 + 0.519i)8-s + (−0.525 + 0.850i)9-s + (1.63 − 0.640i)10-s + (0.247 − 0.202i)11-s + (−0.784 + 0.619i)12-s + (0.814 − 0.435i)13-s + (0.443 − 0.350i)14-s + (0.0314 + 1.75i)15-s + (−0.949 + 0.313i)16-s + (−0.690 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0669173 + 0.116367i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0669173 + 0.116367i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.917 - 1.07i)T \) |
| 3 | \( 1 + (0.843 + 1.51i)T \) |
good | 5 | \( 1 + (3.45 + 1.84i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (1.46 + 0.291i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.819 + 0.672i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.93 + 1.57i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (2.84 - 1.17i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.56 - 5.16i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.50 - 3.01i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (4.19 + 3.44i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (2.35 - 2.35i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.347 + 1.14i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-2.55 - 1.70i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.727 - 7.38i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (9.75 - 4.03i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-6.48 + 5.32i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (7.05 + 3.77i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (1.12 + 11.3i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-8.80 + 0.867i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-1.42 - 0.284i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.57 - 12.9i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (4.74 - 11.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (4.80 - 15.8i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (4.92 + 7.37i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (9.38 + 9.38i)T + 97iT^{2} \) |
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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
−11.34055364929055593771865585624, −11.07765873345614799269460880705, −9.567473821346768174964933191156, −8.364991484419945726037895688993, −8.082832401532832186721466941350, −7.04757283994390885388948684439, −6.16538506167929256392517508567, −5.08122880941377769121476981893, −3.76273975436636024702102366242, −1.26212549166864166967045688651,
0.13313785885205710735699086113, 2.92214611267235160310351553577, 3.79088331154436354414075491539, 4.60206038179055822658474575923, 6.58865397089058593312659700901, 7.25262774388537773099250559406, 8.683277357942846621437972010518, 9.162722703574224801176813805883, 10.44630494496465196795582978236, 11.07821726456291330925182638612