| L(s) = 1 | + (−0.691 − 1.23i)2-s + (−0.471 − 0.881i)3-s + (−1.04 + 1.70i)4-s + (−1.72 + 1.41i)5-s + (−0.762 + 1.19i)6-s + (−0.0630 + 0.0421i)7-s + (2.82 + 0.110i)8-s + (−0.555 + 0.831i)9-s + (2.93 + 1.14i)10-s + (1.01 + 3.35i)11-s + (1.99 + 0.117i)12-s + (4.28 − 5.22i)13-s + (0.0955 + 0.0486i)14-s + (2.05 + 0.853i)15-s + (−1.81 − 3.56i)16-s + (2.61 − 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 − 0.872i)2-s + (−0.272 − 0.509i)3-s + (−0.522 + 0.852i)4-s + (−0.770 + 0.632i)5-s + (−0.311 + 0.486i)6-s + (−0.0238 + 0.0159i)7-s + (0.999 + 0.0391i)8-s + (−0.185 + 0.277i)9-s + (0.928 + 0.363i)10-s + (0.307 + 1.01i)11-s + (0.576 + 0.0339i)12-s + (1.18 − 1.44i)13-s + (0.0255 + 0.0130i)14-s + (0.531 + 0.220i)15-s + (−0.454 − 0.890i)16-s + (0.633 − 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.788478 - 0.330971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.788478 - 0.330971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.691 + 1.23i)T \) |
| 3 | \( 1 + (0.471 + 0.881i)T \) |
| good | 5 | \( 1 + (1.72 - 1.41i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (0.0630 - 0.0421i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.01 - 3.35i)T + (-9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-4.28 + 5.22i)T + (-2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.61 + 1.08i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-5.11 + 0.503i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (5.17 - 1.02i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-6.63 - 2.01i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.28 - 6.28i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.11 + 11.3i)T + (-36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.13 - 5.72i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.45 - 2.72i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (2.54 + 6.13i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-6.60 + 2.00i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (2.05 + 2.50i)T + (-11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-7.68 + 4.10i)T + (33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (3.37 - 1.80i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (0.353 + 0.528i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-12.9 - 8.64i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.108 + 0.262i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.880 - 8.93i)T + (-81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-4.54 - 0.904i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (7.86 + 7.86i)T + 97iT^{2} \) |
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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
−11.25227663893542990190560655078, −10.44237440848744289196954638389, −9.655832834668607729327274661285, −8.263827584296059194568014791744, −7.72469243390985292335876457627, −6.76997741824190291346772779065, −5.27302153858007401025051130595, −3.80056210286543745091820048107, −2.87344427964667288148135817446, −1.10762493024200163670326821209,
0.984516175089289730269857134179, 3.78403803450848398918277115955, 4.58971061562752004447029202404, 5.87183260953228924001691658697, 6.56876938797945630513012654387, 8.065126645480441933525497947355, 8.483259228253038911618519047366, 9.464816532510955002839317692873, 10.33145482022623972421205863574, 11.55267514520724963296839177654
