| L(s) = 1 | + (−0.0564 − 1.41i)2-s + (0.471 − 0.881i)3-s + (−1.99 + 0.159i)4-s + (2.04 + 1.67i)5-s + (−1.27 − 0.616i)6-s + (1.08 + 0.724i)7-s + (0.338 + 2.80i)8-s + (−0.555 − 0.831i)9-s + (2.25 − 2.98i)10-s + (1.60 − 5.27i)11-s + (−0.799 + 1.83i)12-s + (−2.87 − 3.50i)13-s + (0.962 − 1.57i)14-s + (2.44 − 1.01i)15-s + (3.94 − 0.636i)16-s + (4.45 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.0399 − 0.999i)2-s + (0.272 − 0.509i)3-s + (−0.996 + 0.0797i)4-s + (0.913 + 0.749i)5-s + (−0.519 − 0.251i)6-s + (0.409 + 0.273i)7-s + (0.119 + 0.992i)8-s + (−0.185 − 0.277i)9-s + (0.712 − 0.943i)10-s + (0.482 − 1.59i)11-s + (−0.230 + 0.529i)12-s + (−0.797 − 0.971i)13-s + (0.257 − 0.420i)14-s + (0.630 − 0.261i)15-s + (0.987 − 0.159i)16-s + (1.08 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0640 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0640 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08356 - 1.15533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08356 - 1.15533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0564 + 1.41i)T \) |
| 3 | \( 1 + (-0.471 + 0.881i)T \) |
| good | 5 | \( 1 + (-2.04 - 1.67i)T + (0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.08 - 0.724i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.60 + 5.27i)T + (-9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.87 + 3.50i)T + (-2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.45 - 1.84i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-6.82 - 0.672i)T + (18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (3.42 + 0.680i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (9.93 - 3.01i)T + (24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-3.66 + 3.66i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.330 - 3.35i)T + (-36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.60 - 8.07i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-1.63 - 3.04i)T + (-23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (2.83 - 6.85i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (1.86 + 0.566i)T + (44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-1.57 + 1.92i)T + (-11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-11.8 - 6.31i)T + (33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.14 + 0.614i)T + (37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.72 - 7.07i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (3.45 - 2.31i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (1.14 + 2.77i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.465 + 4.72i)T + (-81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (12.8 - 2.56i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-0.0737 + 0.0737i)T - 97iT^{2} \) |
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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
−11.25730613405668903301447217437, −10.09951491400977972621871589035, −9.606448218528523283870300122539, −8.368333893571899632971324221864, −7.65688359021635897501526120815, −6.00492455931944844043330208072, −5.41073268882420941753870946349, −3.46152041343664352230527764257, −2.71954912410914661755511957065, −1.29170208887250597413845134263,
1.73761599758461953614541981456, 3.95455880698838194736301279943, 4.95739398407583787409071753157, 5.53553393511640861128472107134, 7.08036971581599537297991363218, 7.65793931451951008099232232005, 9.036603199277192744449032523500, 9.659733939458646325549946843828, 9.956747357334379037770229318643, 11.77464264241823200162230886984
