L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.690 − 2.12i)5-s − 3·7-s + (0.309 − 0.951i)8-s + (−0.690 + 2.12i)10-s + (−0.190 − 0.138i)11-s + (−0.809 + 0.587i)13-s + (2.42 + 1.76i)14-s + (−0.809 + 0.587i)16-s + (−2.42 + 7.46i)17-s + (−0.263 + 0.812i)19-s + (1.80 − 1.31i)20-s + (0.0729 + 0.224i)22-s + (−5.04 − 3.66i)23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.309 − 0.951i)5-s − 1.13·7-s + (0.109 − 0.336i)8-s + (−0.218 + 0.672i)10-s + (−0.0575 − 0.0418i)11-s + (−0.224 + 0.163i)13-s + (0.648 + 0.471i)14-s + (−0.202 + 0.146i)16-s + (−0.588 + 1.81i)17-s + (−0.0605 + 0.186i)19-s + (0.404 − 0.293i)20-s + (0.0155 + 0.0478i)22-s + (−1.05 − 0.764i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.690 + 2.12i)T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + (0.190 + 0.138i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.42 - 7.46i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (5.04 + 3.66i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.163 - 0.502i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 4.02i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (5.92 - 4.30i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.04 - 4.39i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + (1.83 + 5.65i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.472 + 1.45i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.61 + 2.62i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (1.78 - 5.48i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.927 - 2.85i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 3.35i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.854 + 2.62i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.16 + 12.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.61 + 2.62i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.26 + 10.0i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.26328941547146377056358318445, −9.782259874174138474562083181236, −8.609467959933360660663665116311, −8.230378387629925224543962587662, −6.83841495490255345707359449930, −5.92830136798524843262160322659, −4.41762481619095627401582284407, −3.48093925008119709084154433439, −1.86160265433087180604029033674, 0,
2.49689770459919310804008872713, 3.58858543393454466736065080956, 5.17859677913348361385269910383, 6.42718425893663748905887828693, 7.00440801349771244598839350277, 7.82348501493584776535769314511, 9.100248729028358450552680113994, 9.795218599706978399167678000406, 10.54882286473582133547660390781