L(s) = 1 | + (−0.717 + 1.24i)3-s + (−2.27 + 1.31i)5-s + (5.39 − 4.46i)7-s + (3.47 + 6.01i)9-s + (−2.75 + 4.77i)11-s + 6.22i·13-s − 3.76i·15-s + (−10.5 + 18.2i)17-s + (12.6 + 21.8i)19-s + (1.67 + 9.89i)21-s + (−12.5 + 7.24i)23-s + (−9.05 + 15.6i)25-s − 22.8·27-s − 29.9i·29-s + (23.5 + 13.5i)31-s + ⋯ |
L(s) = 1 | + (−0.239 + 0.414i)3-s + (−0.454 + 0.262i)5-s + (0.770 − 0.637i)7-s + (0.385 + 0.668i)9-s + (−0.250 + 0.433i)11-s + 0.478i·13-s − 0.250i·15-s + (−0.619 + 1.07i)17-s + (0.664 + 1.15i)19-s + (0.0799 + 0.471i)21-s + (−0.545 + 0.314i)23-s + (−0.362 + 0.627i)25-s − 0.846·27-s − 1.03i·29-s + (0.758 + 0.438i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00411 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00411 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.899227 + 0.895535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899227 + 0.895535i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5.39 + 4.46i)T \) |
good | 3 | \( 1 + (0.717 - 1.24i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (2.27 - 1.31i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (2.75 - 4.77i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 6.22iT - 169T^{2} \) |
| 17 | \( 1 + (10.5 - 18.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.6 - 21.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (12.5 - 7.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 29.9iT - 841T^{2} \) |
| 31 | \( 1 + (-23.5 - 13.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-31.5 + 18.2i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 37.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (51.8 - 29.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-36.5 - 21.0i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-14.6 + 25.3i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (92.2 - 53.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.4 + 97.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 94.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-32.3 + 56.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-46.8 + 27.0i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 91.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + (36.1 + 62.5i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 97.4T + 9.40e3T^{2} \) |
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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
−12.08041816331417772620068339524, −11.15116466856233566711675101230, −10.49107932789557167645270153857, −9.563708297230883912540417032581, −7.973654675752256082228875407338, −7.55049866118099780788127024803, −6.03794947362539072355740535037, −4.66008220684750700748968966355, −3.89934490577362362251534907628, −1.81532487332953749779403490281,
0.75185866039655683022941711910, 2.67638798329395471783090056871, 4.39755750995367745693708303688, 5.49532211264667808691710556760, 6.73783199493196747106474064305, 7.80568674381536186522867633184, 8.731051197187260026484431339066, 9.742409480862563926257286886438, 11.20273032308777354107969476255, 11.72602424327782486762586776956