L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + 7i·7-s + (−4 − 6.92i)9-s + (14.7 + 8.5i)11-s + 24·13-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−6.06 + 3.5i)19-s + (3.5 − 6.06i)21-s + (6.06 − 3.5i)23-s + (12 − 20.7i)25-s + 17i·27-s + 24·29-s + (35.5 + 20.5i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.166i)3-s + (−0.100 − 0.173i)5-s + i·7-s + (−0.444 − 0.769i)9-s + (1.33 + 0.772i)11-s + 1.84·13-s + 0.0666i·15-s + (−0.0294 + 0.0509i)17-s + (−0.319 + 0.184i)19-s + (0.166 − 0.288i)21-s + (0.263 − 0.152i)23-s + (0.479 − 0.831i)25-s + 0.629i·27-s + 0.827·29-s + (1.14 + 0.661i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.49772 + 0.149081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49772 + 0.149081i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 3 | \( 1 + (0.866 + 0.5i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-14.7 - 8.5i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 24T + 169T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6.06 - 3.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.06 + 3.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 24T + 841T^{2} \) |
| 31 | \( 1 + (-35.5 - 20.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-24.5 - 42.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + 24iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (47.6 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-12.5 + 21.6i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (14.7 + 8.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (56.2 + 32.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 96iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (47.5 - 82.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (35.5 - 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 72iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (47.5 + 82.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 144T + 9.40e3T^{2} \) |
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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
−11.95931408588974787273697511962, −11.44122853950036505022272838182, −10.07783422942464561078495221759, −8.852471712817776889468075022307, −8.480687416624560540349841719584, −6.52037659626254115318793603819, −6.21939568193788991521427275247, −4.62780547793922233015425645479, −3.26141490713238760708481507861, −1.33638206571630237313296692738,
1.12026154097295026105710592308, 3.36540246700315755760902864489, 4.39927744703282405432001409888, 5.92970404734542269380602983807, 6.76453389539697833731534766449, 8.124221418843820805021395188875, 8.943954435369567798588304496331, 10.31423194653234954216483809557, 11.15639124984579661411175612411, 11.55731188619678256431839426369