| L(s) = 1 | + (0.161 − 1.40i)2-s + (−1.65 + 0.414i)3-s + (−1.94 − 0.453i)4-s + (0.608 + 0.287i)5-s + (0.315 + 2.39i)6-s + (−1.02 − 3.37i)7-s + (−0.951 + 2.66i)8-s + (−0.0776 + 0.0415i)9-s + (0.502 − 0.808i)10-s + (−3.45 + 0.512i)11-s + (3.41 − 0.0567i)12-s + (−1.81 + 5.07i)13-s + (−4.90 + 0.892i)14-s + (−1.12 − 0.224i)15-s + (3.58 + 1.76i)16-s + (−2.95 + 0.587i)17-s + ⋯ |
| L(s) = 1 | + (0.114 − 0.993i)2-s + (−0.955 + 0.239i)3-s + (−0.973 − 0.226i)4-s + (0.272 + 0.128i)5-s + (0.128 + 0.976i)6-s + (−0.386 − 1.27i)7-s + (−0.336 + 0.941i)8-s + (−0.0258 + 0.0138i)9-s + (0.158 − 0.255i)10-s + (−1.04 + 0.154i)11-s + (0.985 − 0.0163i)12-s + (−0.503 + 1.40i)13-s + (−1.31 + 0.238i)14-s + (−0.290 − 0.0578i)15-s + (0.897 + 0.441i)16-s + (−0.716 + 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0445718 + 0.0933807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0445718 + 0.0933807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.161 + 1.40i)T \) |
| good | 3 | \( 1 + (1.65 - 0.414i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-0.608 - 0.287i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (1.02 + 3.37i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (3.45 - 0.512i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (1.81 - 5.07i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (2.95 - 0.587i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (1.32 + 1.20i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.382 - 0.0376i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-0.228 - 0.307i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (2.67 + 1.10i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.180 + 3.68i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-1.61 + 1.96i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-0.249 + 0.996i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (5.05 + 3.37i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-6.43 + 8.68i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-3.30 - 9.23i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (6.16 - 10.2i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (9.37 + 5.62i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (5.74 - 10.7i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-4.38 + 14.4i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (6.22 + 9.32i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.681 - 13.8i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-9.89 - 0.974i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 16.1i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.29545234208692565380096079310, −10.50932149243421979016734436385, −10.02717838673648674852524459743, −8.825202706922190518760533115924, −7.33161487257276078423127684528, −6.12425659920969612461932633282, −4.87213705210305084605793987378, −4.05680496521926549224897147797, −2.29229401873609251293372250280, −0.082647760621158035380563383428,
2.94699958283328578975417042203, 5.03452359391102550816209082190, 5.63936940770311043906938741928, 6.31178154095465194024853806499, 7.64047709183002925500649334654, 8.605042829774027669429097647110, 9.602636763557271865019407219589, 10.70130282224642113234720490632, 11.95923239263385527398369222731, 12.76384997318581058419625011267
