L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.0279 + 0.194i)5-s + (0.841 − 0.540i)6-s + (−0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.0816 − 0.178i)10-s + (−0.219 + 0.0644i)11-s + (−0.959 + 0.281i)12-s + (−1.17 + 2.57i)13-s + (0.142 + 0.989i)14-s + (−0.128 − 0.148i)15-s + (0.415 + 0.909i)16-s + (5.79 − 3.72i)17-s + ⋯ |
L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0125 + 0.0869i)5-s + (0.343 − 0.220i)6-s + (−0.157 − 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.0258 − 0.0565i)10-s + (−0.0661 + 0.0194i)11-s + (−0.276 + 0.0813i)12-s + (−0.325 + 0.713i)13-s + (0.0380 + 0.264i)14-s + (−0.0332 − 0.0383i)15-s + (0.103 + 0.227i)16-s + (1.40 − 0.902i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.894759 - 0.186499i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.894759 - 0.186499i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.96 + 4.37i)T \) |
good | 5 | \( 1 + (0.0279 - 0.194i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.219 - 0.0644i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.17 - 2.57i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.79 + 3.72i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.47 + 1.58i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.42 - 0.918i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.220 - 0.254i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.806 - 5.60i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.78 + 12.4i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.52 + 1.76i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 + (-0.831 - 1.82i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.58 + 3.46i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-7.21 - 8.32i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-7.19 - 2.11i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-13.2 - 3.90i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.82 - 3.09i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.61 + 5.72i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (2.21 + 15.4i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (1.06 - 1.23i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.60 + 18.1i)T + (-93.0 - 27.3i)T^{2} \) |
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
−10.05218194908862239140265726448, −9.242035284162523662315790230600, −8.502120819120898485959934500082, −7.34878501947403240092315421768, −6.80285791342353302126446981864, −5.66005325143244033772280185986, −4.63657121574628275775928998444, −3.57136389665980644585376514828, −2.39697800303820445889979368712, −0.72432196067907810823853674459,
1.01127440512453681618983232428, 2.34500633713620153627490877293, 3.64365649494030998048369969043, 5.19293154243780591376085135786, 5.88115148597538379751774951923, 6.68251822731283206826389516220, 7.87819610870853117329455981656, 8.071573144314606150399995298270, 9.297143236400055694713158523522, 10.04178909636262293388010576827