| L(s) = 1 | − i·3-s − 4.02i·5-s − 4.61·7-s − 9-s − 1.53i·11-s − 5.57i·13-s − 4.02·15-s − 1.29·17-s + 4.35i·19-s + 4.61i·21-s − 4·23-s − 11.2·25-s + i·27-s − 1.86i·29-s − 2.77·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.80i·5-s − 1.74·7-s − 0.333·9-s − 0.461i·11-s − 1.54i·13-s − 1.03·15-s − 0.314·17-s + 1.00i·19-s + 1.00i·21-s − 0.834·23-s − 2.24·25-s + 0.192i·27-s − 0.345i·29-s − 0.498·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4596242980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4596242980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 4.02iT - 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.53iT - 11T^{2} \) |
| 13 | \( 1 + 5.57iT - 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 4.35iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.86iT - 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 + 3.03iT - 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 5.73iT - 59T^{2} \) |
| 61 | \( 1 + 7.64iT - 61T^{2} \) |
| 67 | \( 1 + 8.79iT - 67T^{2} \) |
| 71 | \( 1 - 6.88T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 - 4.12iT - 83T^{2} \) |
| 89 | \( 1 + 7.98T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.091310112693451839372719882903, −7.65470057586770308280698237814, −6.44189914048043045139651204373, −5.77833059471777704913064260723, −5.37829290363689071185665483126, −4.06930305458799478611451031247, −3.40406721412281381076822430234, −2.24407821694239672562934071038, −0.892732325963023267386936636844, −0.17498694465784485072700249497,
2.25362164516126077868187794983, 2.86286146525259832054211223741, 3.72613924156474986441859313905, 4.30124230482591793653857164960, 5.69741053677613411175702014489, 6.52576978899869718858638039897, 6.80970433146076511072073675020, 7.42871182839624498587608067816, 8.791168460233105055016726764975, 9.496369474337811869954353906028
