L(s) = 1 | + (0.5 + 0.866i)2-s + (0.278 − 1.70i)3-s + (−0.499 + 0.866i)4-s + (2.44 − 1.41i)5-s + (1.61 − 0.613i)6-s + (−2.60 + 0.449i)7-s − 0.999·8-s + (−2.84 − 0.950i)9-s + (2.44 + 1.41i)10-s + (2.97 − 5.15i)11-s + (1.34 + 1.09i)12-s + (3.07 − 1.87i)13-s + (−1.69 − 2.03i)14-s + (−1.73 − 4.58i)15-s + (−0.5 − 0.866i)16-s + (−2.83 + 4.91i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.160 − 0.987i)3-s + (−0.249 + 0.433i)4-s + (1.09 − 0.632i)5-s + (0.661 − 0.250i)6-s + (−0.985 + 0.169i)7-s − 0.353·8-s + (−0.948 − 0.316i)9-s + (0.774 + 0.447i)10-s + (0.897 − 1.55i)11-s + (0.387 + 0.316i)12-s + (0.854 − 0.520i)13-s + (−0.452 − 0.543i)14-s + (−0.448 − 1.18i)15-s + (−0.125 − 0.216i)16-s + (−0.688 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70932 - 0.812499i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70932 - 0.812499i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.278 + 1.70i)T \) |
| 7 | \( 1 + (2.60 - 0.449i)T \) |
| 13 | \( 1 + (-3.07 + 1.87i)T \) |
good | 5 | \( 1 + (-2.44 + 1.41i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 5.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.972 + 1.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.43 + 1.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.0658iT - 29T^{2} \) |
| 31 | \( 1 + (-4.39 + 7.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.160 + 0.0928i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.04iT - 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + (6.77 - 3.91i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.77 - 2.18i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.6 - 6.71i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12.8 - 7.39i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.912 - 0.527i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.16T + 71T^{2} \) |
| 73 | \( 1 + (0.140 - 0.242i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 7.24i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.940iT - 83T^{2} \) |
| 89 | \( 1 + (4.61 - 2.66i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−10.76805904390048817534255923755, −9.356552566657421709921308936003, −8.784289050152357883407358962512, −8.136195952902602767163930150793, −6.59336427962424840518599306367, −6.16265751244326137837132884231, −5.64830197538126770048613099096, −3.87638172881005430679465254377, −2.70142851663622711911766630385, −1.03068281958651510851577592870,
2.00565501335496533688645094367, 3.11214769167932381940650716766, 4.12183367247891742148166027910, 5.11253740494932133932960993831, 6.35298169028943091291013742478, 6.90330426453219022715047916757, 8.862262935213380219052248987703, 9.530640684880265218771625704131, 9.962440808163258592589099760043, 10.73367806110587508372246040534