L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s − 2.82i·5-s − 1.73i·7-s + 2.82i·8-s + (2.00 + 3.46i)10-s + 4.89·11-s − 13-s + (1.22 + 2.12i)14-s + (−2.00 − 3.46i)16-s − 2.82i·17-s + 5.19i·19-s + (−4.89 − 2.82i)20-s + (−5.99 + 3.46i)22-s − 4.89·23-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)2-s + (0.499 − 0.866i)4-s − 1.26i·5-s − 0.654i·7-s + 0.999i·8-s + (0.632 + 1.09i)10-s + 1.47·11-s − 0.277·13-s + (0.327 + 0.566i)14-s + (−0.500 − 0.866i)16-s − 0.685i·17-s + 1.19i·19-s + (−1.09 − 0.632i)20-s + (−1.27 + 0.738i)22-s − 1.02·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.702105 - 0.188128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.702105 - 0.188128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−13.88899356930377682860245414798, −12.41742676422328489781301743072, −11.51030860027074266517156018103, −10.07056945536151663804908059052, −9.231739868664361363571273147442, −8.285813732700835825090870302135, −7.10773983113878985981094762058, −5.77710144178911363071619859598, −4.31239716399917921215721847652, −1.28738494146991810369837546020,
2.31677886908070896022890697102, 3.78108429902194902533382754795, 6.29241448203311121119305129276, 7.16132092439863928675358722446, 8.581188624764473361244357411680, 9.579514382579032216601020781947, 10.63309385102304075552945923961, 11.56634646519630827357679526066, 12.31226909877379339742711803629, 13.84615297786431326546519009863