L(s) = 1 | + (1.40 + 0.166i)2-s + (−1.62 + 0.593i)3-s + (1.94 + 0.466i)4-s + (0.720 + 1.07i)5-s + (−2.38 + 0.563i)6-s + (−0.767 + 1.85i)7-s + (2.65 + 0.978i)8-s + (2.29 − 1.93i)9-s + (0.833 + 1.63i)10-s + (−0.967 − 0.192i)11-s + (−3.44 + 0.395i)12-s + (2.76 + 1.84i)13-s + (−1.38 + 2.47i)14-s + (−1.81 − 1.32i)15-s + (3.56 + 1.81i)16-s + (−4.20 − 4.20i)17-s + ⋯ |
L(s) = 1 | + (0.993 + 0.117i)2-s + (−0.939 + 0.342i)3-s + (0.972 + 0.233i)4-s + (0.322 + 0.482i)5-s + (−0.973 + 0.229i)6-s + (−0.289 + 0.699i)7-s + (0.938 + 0.345i)8-s + (0.765 − 0.643i)9-s + (0.263 + 0.516i)10-s + (−0.291 − 0.0580i)11-s + (−0.993 + 0.114i)12-s + (0.767 + 0.512i)13-s + (−0.370 + 0.661i)14-s + (−0.468 − 0.342i)15-s + (0.891 + 0.453i)16-s + (−1.01 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48969 + 0.737815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48969 + 0.737815i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.166i)T \) |
| 3 | \( 1 + (1.62 - 0.593i)T \) |
good | 5 | \( 1 + (-0.720 - 1.07i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.767 - 1.85i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.967 + 0.192i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 1.84i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (4.20 + 4.20i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.31 - 0.879i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (3.41 + 8.25i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.303 - 1.52i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 + (3.53 + 5.29i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (0.606 - 0.251i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (7.79 + 1.54i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-3.56 + 3.56i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.605 + 3.04i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (6.01 - 4.01i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.31 - 11.6i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-12.2 + 2.43i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-10.2 - 4.23i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (8.22 - 3.40i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-4.22 + 4.22i)T - 79iT^{2} \) |
| 83 | \( 1 + (8.63 - 12.9i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (10.4 + 4.32i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 4.02iT - 97T^{2} \) |
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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
−12.55268449759154834395220834793, −11.81692218556460653933829662755, −10.93956893941462348078806613841, −10.12542939820177143128114088157, −8.690372640906882708119334375575, −6.90307549859749017459702357656, −6.30162787720886154846412302201, −5.27840139971959648657240180234, −4.12377871791821160518405059750, −2.50863543334676837693087700900,
1.54811877853175509554612031903, 3.70880129500629187579853141200, 4.97931698797219112214821618700, 5.93099687173723650305129885509, 6.84410689260906250679827416248, 8.003998255621574504767296000788, 9.851092067198715640923864564087, 10.76347830799372217109012397325, 11.49758145867779592028254966527, 12.59971552982382744731560708388