L(s) = 1 | + (−1.11 + 1.93i)2-s + (0.5 + 0.866i)3-s + (−1.5 − 2.59i)4-s + (0.5 − 0.866i)5-s − 2.23·6-s + 2.23·8-s + (−0.499 + 0.866i)9-s + (1.11 + 1.93i)10-s + (1.23 + 2.14i)11-s + (1.50 − 2.59i)12-s − 4.47·13-s + 0.999·15-s + (0.499 − 0.866i)16-s + (1 + 1.73i)17-s + (−1.11 − 1.93i)18-s + (−3.23 + 5.60i)19-s + ⋯ |
L(s) = 1 | + (−0.790 + 1.36i)2-s + (0.288 + 0.499i)3-s + (−0.750 − 1.29i)4-s + (0.223 − 0.387i)5-s − 0.912·6-s + 0.790·8-s + (−0.166 + 0.288i)9-s + (0.353 + 0.612i)10-s + (0.372 + 0.645i)11-s + (0.433 − 0.749i)12-s − 1.24·13-s + 0.258·15-s + (0.124 − 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.263 − 0.456i)18-s + (−0.742 + 1.28i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.237752 - 0.479602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.237752 - 0.479602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.11 - 1.93i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.23 - 5.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (5.23 + 9.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.47 - 9.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + (-2.47 + 4.28i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.23 - 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.47 + 7.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (-1.76 - 3.05i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.47 + 4.28i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.944T + 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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
−10.28748356085444178849188995109, −9.809507850673465420052567399260, −9.184753909315392548937122914065, −8.218648032064520578563075607625, −7.67058314773008364313889522775, −6.69944583373681377336154761971, −5.73072100796744867386152415230, −4.93682640775449739583981812286, −3.76406455234968437947791294648, −1.92521740391987320148288821105,
0.32659915608370331356314198235, 1.91464459574307210017226161144, 2.72787487584837948335430939110, 3.69236108011718957281815850222, 5.17141695414646806116174598558, 6.57717233494065938225670977925, 7.35507989455265822257232435521, 8.534364846864236732170055215138, 9.054158095291252054087391934897, 9.904065673761121605249884107176