| 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 | \( 1 + (-2.60 - 18.1i)T + (-93.0 + 27.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04178909636262293388010576827, −9.297143236400055694713158523522, −8.071573144314606150399995298270, −7.87819610870853117329455981656, −6.68251822731283206826389516220, −5.88115148597538379751774951923, −5.19293154243780591376085135786, −3.64365649494030998048369969043, −2.34500633713620153627490877293, −1.01127440512453681618983232428,
0.72432196067907810823853674459, 2.39697800303820445889979368712, 3.57136389665980644585376514828, 4.63657121574628275775928998444, 5.66005325143244033772280185986, 6.80285791342353302126446981864, 7.34878501947403240092315421768, 8.502120819120898485959934500082, 9.242035284162523662315790230600, 10.05218194908862239140265726448
