L(s) = 1 | − 2i·2-s − i·3-s − 2·4-s − 2·6-s + 3i·7-s − 9-s + 2·11-s + 2i·12-s + i·13-s + 6·14-s − 4·16-s − 2i·17-s + 2i·18-s + 5·19-s + 3·21-s − 4i·22-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 0.577i·3-s − 4-s − 0.816·6-s + 1.13i·7-s − 0.333·9-s + 0.603·11-s + 0.577i·12-s + 0.277i·13-s + 1.60·14-s − 16-s − 0.485i·17-s + 0.471i·18-s + 1.14·19-s + 0.654·21-s − 0.852i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.490983 - 0.794428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.490983 - 0.794428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.76560097149990409305510756326, −12.82684476259162027079655811408, −11.74278915826813398513090580853, −11.44560299852911851411715448857, −9.715877541712923973372115428251, −8.942075476039868462130469896912, −7.19132413134035066790980104844, −5.50602029648065362711011946692, −3.42992584500941121446531627913, −1.87406895516215122089339214148,
3.95467184940752398421650199006, 5.36869590813685455208628735030, 6.73196839614468195283312946179, 7.75232414789649101971239372127, 8.998851506099230354236081802260, 10.28050345487040540000848756328, 11.47012906506001366604380880864, 13.21515765633412552018180043720, 14.26732083822925441082468851828, 14.91080771710218934571399627491