L(s) = 1 | + (−1.39 + 0.221i)2-s + 2.04i·3-s + (1.90 − 0.619i)4-s + (0.665 − 2.13i)5-s + (−0.453 − 2.85i)6-s − 2.07·7-s + (−2.51 + 1.28i)8-s − 1.17·9-s + (−0.456 + 3.12i)10-s + 4.08i·11-s + (1.26 + 3.88i)12-s + 4.52·13-s + (2.90 − 0.460i)14-s + (4.36 + 1.36i)15-s + (3.23 − 2.35i)16-s + 6.58i·17-s + ⋯ |
L(s) = 1 | + (−0.987 + 0.156i)2-s + 1.17i·3-s + (0.950 − 0.309i)4-s + (0.297 − 0.954i)5-s + (−0.184 − 1.16i)6-s − 0.785·7-s + (−0.890 + 0.454i)8-s − 0.392·9-s + (−0.144 + 0.989i)10-s + 1.23i·11-s + (0.365 + 1.12i)12-s + 1.25·13-s + (0.775 − 0.123i)14-s + (1.12 + 0.351i)15-s + (0.808 − 0.588i)16-s + 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.547573 + 0.635621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.547573 + 0.635621i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.221i)T \) |
| 5 | \( 1 + (-0.665 + 2.13i)T \) |
| 19 | \( 1 + (4.31 - 0.590i)T \) |
good | 3 | \( 1 - 2.04iT - 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 - 4.08iT - 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 6.58iT - 17T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 5.90iT - 29T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 1.10iT - 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 + 2.18T + 53T^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 4.67T + 71T^{2} \) |
| 73 | \( 1 - 6.18iT - 73T^{2} \) |
| 79 | \( 1 - 4.04T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 + 0.553iT - 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−11.10832488816279478816486455421, −10.40374882929718977965969420610, −9.739391975373594920515321481659, −8.960106159703109663125045367708, −8.377513479597299526702860751031, −6.87512714376351077676289557479, −5.92277037015308224210018018958, −4.71580416699268713157616413033, −3.55283876929683632343084949393, −1.62857871803117528094463514402,
0.814327854817356749995981947976, 2.47398650605596737403071763659, 3.35662495320450277030571640698, 6.01184932528531586525411814338, 6.55981404045785618144699522717, 7.25478481813478781020398517070, 8.304236582130996339179571718712, 9.195038577298561948494205676816, 10.21416578614197027227943897152, 11.20477077178336351528630261444