| L(s) = 1 | + (−1.37 + 0.346i)2-s + (−0.740 + 0.671i)3-s + (1.76 − 0.949i)4-s + (0.359 + 1.43i)5-s + (0.783 − 1.17i)6-s + (2.14 + 1.76i)7-s + (−2.08 + 1.91i)8-s + (0.0980 − 0.995i)9-s + (−0.989 − 1.84i)10-s + (−2.92 + 1.38i)11-s + (−0.666 + 1.88i)12-s + (−1.91 + 1.15i)13-s + (−3.55 − 1.67i)14-s + (−1.22 − 0.821i)15-s + (2.19 − 3.34i)16-s + (−2.64 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.969 + 0.244i)2-s + (−0.427 + 0.387i)3-s + (0.880 − 0.474i)4-s + (0.160 + 0.641i)5-s + (0.319 − 0.480i)6-s + (0.811 + 0.665i)7-s + (−0.737 + 0.675i)8-s + (0.0326 − 0.331i)9-s + (−0.312 − 0.582i)10-s + (−0.883 + 0.417i)11-s + (−0.192 + 0.544i)12-s + (−0.532 + 0.319i)13-s + (−0.949 − 0.446i)14-s + (−0.317 − 0.212i)15-s + (0.549 − 0.835i)16-s + (−0.640 + 0.428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0432828 + 0.552404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0432828 + 0.552404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.346i)T \) |
| 3 | \( 1 + (0.740 - 0.671i)T \) |
| good | 5 | \( 1 + (-0.359 - 1.43i)T + (-4.40 + 2.35i)T^{2} \) |
| 7 | \( 1 + (-2.14 - 1.76i)T + (1.36 + 6.86i)T^{2} \) |
| 11 | \( 1 + (2.92 - 1.38i)T + (6.97 - 8.50i)T^{2} \) |
| 13 | \( 1 + (1.91 - 1.15i)T + (6.12 - 11.4i)T^{2} \) |
| 17 | \( 1 + (2.64 - 1.76i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (1.15 - 1.56i)T + (-5.51 - 18.1i)T^{2} \) |
| 23 | \( 1 + (-3.61 + 1.09i)T + (19.1 - 12.7i)T^{2} \) |
| 29 | \( 1 + (3.19 - 1.14i)T + (22.4 - 18.3i)T^{2} \) |
| 31 | \( 1 + (-1.75 - 4.23i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.956 + 6.44i)T + (-35.4 - 10.7i)T^{2} \) |
| 41 | \( 1 + (5.30 + 2.83i)T + (22.7 + 34.0i)T^{2} \) |
| 43 | \( 1 + (5.15 - 5.69i)T + (-4.21 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.14 - 5.76i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-1.49 - 0.533i)T + (40.9 + 33.6i)T^{2} \) |
| 59 | \( 1 + (3.43 + 2.05i)T + (27.8 + 52.0i)T^{2} \) |
| 61 | \( 1 + (6.70 + 0.329i)T + (60.7 + 5.97i)T^{2} \) |
| 67 | \( 1 + (-0.0170 + 0.347i)T + (-66.6 - 6.56i)T^{2} \) |
| 71 | \( 1 + (-3.61 + 0.356i)T + (69.6 - 13.8i)T^{2} \) |
| 73 | \( 1 + (9.98 - 8.19i)T + (14.2 - 71.5i)T^{2} \) |
| 79 | \( 1 + (3.58 - 0.714i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-0.558 - 3.76i)T + (-79.4 + 24.0i)T^{2} \) |
| 89 | \( 1 + (12.2 + 3.71i)T + (74.0 + 49.4i)T^{2} \) |
| 97 | \( 1 + (-14.6 + 6.07i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.68401209470094956216158401842, −9.956540336464869615471331293941, −9.011840341010059085780996648704, −8.272923368965982001583912651314, −7.29232288522518235164419563552, −6.51490814515556707941790735802, −5.50427663775930425308835007779, −4.69415033753742622298916961346, −2.86212885751030062691473342995, −1.84898071148632717603279921664,
0.39332955056169862129184332212, 1.63910138216216903600067687565, 2.92162911895982391460067991799, 4.59586390327260365365973141417, 5.44073907811887348756552536189, 6.71548968292947984811830930061, 7.50289390406607782936461341427, 8.206071859375689784368284372696, 8.982612346283204253992872141445, 10.01585662622039892758554492196
