| L(s) = 1 | + (0.358 − 1.36i)2-s + (0.984 − 1.42i)3-s + (−1.74 − 0.980i)4-s + (−0.494 + 1.35i)5-s + (−1.59 − 1.85i)6-s + (−0.984 − 0.173i)7-s + (−1.96 + 2.03i)8-s + (−1.06 − 2.80i)9-s + (1.68 + 1.16i)10-s + (−5.03 + 1.83i)11-s + (−3.11 + 1.51i)12-s + (−0.663 + 0.556i)13-s + (−0.590 + 1.28i)14-s + (1.45 + 2.04i)15-s + (2.07 + 3.41i)16-s + (−1.85 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.253 − 0.967i)2-s + (0.568 − 0.822i)3-s + (−0.871 − 0.490i)4-s + (−0.221 + 0.607i)5-s + (−0.652 − 0.758i)6-s + (−0.372 − 0.0656i)7-s + (−0.694 + 0.719i)8-s + (−0.353 − 0.935i)9-s + (0.532 + 0.368i)10-s + (−1.51 + 0.553i)11-s + (−0.898 + 0.438i)12-s + (−0.184 + 0.154i)13-s + (−0.157 + 0.343i)14-s + (0.374 + 0.527i)15-s + (0.519 + 0.854i)16-s + (−0.449 + 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.251274 + 0.382649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251274 + 0.382649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.358 + 1.36i)T \) |
| 3 | \( 1 + (-0.984 + 1.42i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (0.494 - 1.35i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (5.03 - 1.83i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.663 - 0.556i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.85 - 1.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.55 + 3.78i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.924 + 5.24i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.138 + 0.165i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.62 + 0.638i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.89 + 3.27i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0788 + 0.0939i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.96 + 10.8i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.31 - 7.46i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.19iT - 53T^{2} \) |
| 59 | \( 1 + (-8.22 - 2.99i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 7.68i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.23 - 5.05i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.69 - 2.93i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.904 + 1.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.28 - 2.71i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.20 - 5.20i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.42 + 4.28i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 4.40i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.00113795063560647291068322436, −8.854966496798075848870381043172, −8.240115935360913142278749822614, −7.10676674812210525873848637721, −6.35871827145045293842696212761, −5.02812052221029111363911979680, −3.91202541130420075361004645181, −2.68513295070466177391124686170, −2.22321909669659595955737619282, −0.18427326304392217302048039781,
2.70522031310308402673870415435, 3.77197143856437761215929007693, 4.75711025573839643773518919600, 5.41099600351089722420980045891, 6.46036893468046411024269601857, 7.82995595405739041795974075742, 8.253889396611361374170037723647, 8.960151129289762541040412529037, 9.946514070682976025791665532389, 10.57992269387553077474684150964
