| L(s) = 1 | + (−0.647 + 1.02i)2-s + (0.0629 − 0.0591i)3-s + (0.229 + 0.488i)4-s + (2.06 − 0.853i)5-s + (0.0195 + 0.102i)6-s + (1.93 − 1.40i)7-s + (−3.04 − 0.384i)8-s + (−0.187 + 2.98i)9-s + (−0.467 + 2.66i)10-s + (−1.49 + 2.35i)11-s + (0.0433 + 0.0171i)12-s + (−0.0683 + 1.08i)13-s + (0.181 + 2.88i)14-s + (0.0796 − 0.175i)15-s + (1.67 − 2.02i)16-s + (2.34 − 4.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.457 + 0.721i)2-s + (0.0363 − 0.0341i)3-s + (0.114 + 0.244i)4-s + (0.924 − 0.381i)5-s + (0.00798 + 0.0418i)6-s + (0.732 − 0.532i)7-s + (−1.07 − 0.135i)8-s + (−0.0626 + 0.995i)9-s + (−0.147 + 0.841i)10-s + (−0.450 + 0.709i)11-s + (0.0125 + 0.00495i)12-s + (−0.0189 + 0.301i)13-s + (0.0485 + 0.771i)14-s + (0.0205 − 0.0454i)15-s + (0.418 − 0.506i)16-s + (0.569 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864731 + 0.544978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864731 + 0.544978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.06 + 0.853i)T \) |
| good | 2 | \( 1 + (0.647 - 1.02i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (-0.0629 + 0.0591i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 1.40i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (1.49 - 2.35i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.0683 - 1.08i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (-2.34 + 4.99i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (0.157 + 0.148i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (5.98 + 1.53i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (4.65 + 2.55i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (-1.56 + 3.31i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (0.634 - 0.767i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-8.62 + 2.21i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-1.84 + 5.66i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (9.08 - 1.14i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (1.69 - 8.87i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (11.6 + 4.62i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-10.6 - 2.73i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (-1.99 + 1.09i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (10.0 - 1.26i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (9.83 - 3.89i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-1.08 + 1.01i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-7.46 - 7.01i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-12.1 + 4.81i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (2.02 + 1.11i)T + (51.9 + 81.8i)T^{2} \) |
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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
−13.75121972499000709519757670383, −12.67163275265445978872924995539, −11.51771164152496493002917606212, −10.22766284606006770506531398722, −9.253079359027877578207147674107, −7.968077806691374344662366154674, −7.37055617026776380855301320013, −5.87588611924324196105892841723, −4.65551481366099681457345797314, −2.28440645517057906346934993002,
1.70704100978317803610856231817, 3.21325236536608318011293652868, 5.58625239378305976579124200090, 6.25175830836617166323005107909, 8.188641860209225775691342332692, 9.254129887181912801858853443377, 10.15733324887712322718107640835, 11.00122704254268520620030182423, 11.96197555126893089706766670781, 13.02149146290836932697987234434
