L(s) = 1 | + (0.142 + 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (0.959 + 0.281i)6-s + (−3.23 + 2.07i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.436 + 3.03i)11-s + (−0.142 + 0.989i)12-s + (5.09 + 3.27i)13-s + (−2.51 − 2.90i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (4.68 + 1.37i)17-s + ⋯ |
L(s) = 1 | + (0.100 + 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (0.391 + 0.115i)6-s + (−1.22 + 0.785i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.266 + 0.170i)10-s + (−0.131 + 0.914i)11-s + (−0.0410 + 0.285i)12-s + (1.41 + 0.907i)13-s + (−0.673 − 0.776i)14-s + (−0.107 − 0.234i)15-s + (0.210 − 0.135i)16-s + (1.13 + 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.862387 + 1.03991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.862387 + 1.03991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (4.47 - 1.71i)T \) |
good | 7 | \( 1 + (3.23 - 2.07i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.436 - 3.03i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.09 - 3.27i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.68 - 1.37i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (5.19 - 1.52i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-7.16 - 2.10i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.85 - 8.44i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.683 - 0.788i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (3.48 - 4.02i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.81 + 10.5i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + (2.28 - 1.46i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.57 - 3.57i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.39 + 5.23i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.58 + 11.0i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.89 - 13.2i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (14.7 - 4.34i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (8.79 + 5.65i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.49 - 5.18i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.42 + 3.12i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (0.929 - 1.07i)T + (-13.8 - 96.0i)T^{2} \) |
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
−10.39367415938435768362092047409, −9.686107792205314168794441936339, −8.717565831502753929010322773218, −8.306710505006877131896219196103, −6.94225715763757943692146806377, −6.33686929167967184290601223639, −5.66279735514466514333982730826, −4.28946392179420081761606717592, −3.15441785019129068270486665120, −1.66327490232008426633728409712,
0.69859931894393653015242090965, 2.76136654211701207438465317124, 3.45158040858803401883916881024, 4.30297421412043967016598214698, 5.87862581607393572670805547578, 6.32444079258127489797019662028, 7.88803386616593588847845134188, 8.626763371553946534105797822001, 9.731506600777203637568032430283, 10.28019804047210581551757793005