L(s) = 1 | + (1.57 + 0.724i)3-s + (1.22 − 0.707i)5-s + 0.267·7-s + (1.94 + 2.28i)9-s − 5.27i·11-s + (−0.232 − 0.133i)13-s + (2.43 − 0.224i)15-s + (4.24 − 2.44i)17-s + (−1.73 − 4i)19-s + (0.421 + 0.194i)21-s + (4.57 + 2.63i)23-s + (−1.50 + 2.59i)25-s + (1.41 + 5.00i)27-s + (1.03 − 1.79i)29-s + 2.46i·31-s + ⋯ |
L(s) = 1 | + (0.908 + 0.418i)3-s + (0.547 − 0.316i)5-s + 0.101·7-s + (0.649 + 0.760i)9-s − 1.59i·11-s + (−0.0643 − 0.0371i)13-s + (0.629 − 0.0580i)15-s + (1.02 − 0.594i)17-s + (−0.397 − 0.917i)19-s + (0.0919 + 0.0423i)21-s + (0.953 + 0.550i)23-s + (−0.300 + 0.519i)25-s + (0.272 + 0.962i)27-s + (0.192 − 0.332i)29-s + 0.442i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.45410 - 0.148620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45410 - 0.148620i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 - 0.724i)T \) |
| 19 | \( 1 + (1.73 + 4i)T \) |
good | 5 | \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 0.267T + 7T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + (0.232 + 0.133i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.24 + 2.44i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.57 - 2.63i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.03 + 1.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.46iT - 31T^{2} \) |
| 37 | \( 1 - 7.73iT - 37T^{2} \) |
| 41 | \( 1 + (2.82 + 4.89i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.86 - 4.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.656 - 0.378i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.46 - 9.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.60 + 9.71i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.69 - 11.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.06 - 5.23i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.07iT - 83T^{2} \) |
| 89 | \( 1 + (-3.67 + 6.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.464 - 0.267i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.812383101093650997201855976077, −9.269521333439895806402051770224, −8.494864815700259997744010161777, −7.80242218373494912758222447382, −6.69990489361205828565400423504, −5.53194687821988320707875493814, −4.82883385406660170728445990119, −3.46941770451879592166586638229, −2.79564905164361778775530653270, −1.26208926721434338163630284158,
1.59237011117653558162478786126, 2.42401625082568662914241303228, 3.64093170052076480974374695206, 4.66315205620051863385202513106, 5.95005323001668172215384439535, 6.84580218528106452646758849184, 7.59421506736130961115441457131, 8.352234331672157124279951521365, 9.369595837637582095702968598426, 9.989806266929107579862466424944