| L(s) = 1 | + (0.644 + 1.25i)2-s + (0.836 + 1.51i)3-s + (−1.17 + 1.62i)4-s + (−0.0400 + 0.110i)5-s + (−1.37 + 2.02i)6-s + (−0.984 − 0.173i)7-s + (−2.79 − 0.428i)8-s + (−1.60 + 2.53i)9-s + (−0.164 + 0.0204i)10-s + (−2.20 + 0.801i)11-s + (−3.43 − 0.418i)12-s + (−1.62 + 1.36i)13-s + (−0.415 − 1.35i)14-s + (−0.200 + 0.0312i)15-s + (−1.26 − 3.79i)16-s + (2.23 − 1.28i)17-s + ⋯ |
| L(s) = 1 | + (0.455 + 0.890i)2-s + (0.482 + 0.875i)3-s + (−0.585 + 0.810i)4-s + (−0.0179 + 0.0491i)5-s + (−0.559 + 0.828i)6-s + (−0.372 − 0.0656i)7-s + (−0.988 − 0.151i)8-s + (−0.533 + 0.845i)9-s + (−0.0519 + 0.00646i)10-s + (−0.664 + 0.241i)11-s + (−0.992 − 0.120i)12-s + (−0.450 + 0.377i)13-s + (−0.111 − 0.361i)14-s + (−0.0517 + 0.00807i)15-s + (−0.315 − 0.949i)16-s + (0.541 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.336613 - 1.37660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.336613 - 1.37660i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.644 - 1.25i)T \) |
| 3 | \( 1 + (-0.836 - 1.51i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (0.0400 - 0.110i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (2.20 - 0.801i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.62 - 1.36i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.23 + 1.28i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.48 + 1.43i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.586 - 3.32i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.464 - 0.553i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.32 - 0.410i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.30 - 3.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.60 - 5.49i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.311 + 0.856i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.28 - 12.9i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 3.88iT - 53T^{2} \) |
| 59 | \( 1 + (-5.02 - 1.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.26 + 12.8i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.426 + 0.507i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.17 + 2.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.70 + 6.42i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.67 - 7.95i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.00 - 6.71i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.97 - 2.29i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.17 + 1.88i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.72243154442948781679123212740, −9.617155502807901540697691886435, −9.230688828248020254641976267833, −8.114391640421345072591559321942, −7.47308619422375679718300886155, −6.43204357577694831674037865427, −5.27727362647377669938992902019, −4.67379473614841560604189966008, −3.56314476382484617621500763290, −2.67349157227811052429491696143,
0.56410978744117478456747607180, 2.14665934174847499284193818245, 2.96214186291638893581037575064, 4.02811217028234003918114934843, 5.37830139394735582411999760304, 6.16463154953027374543586511931, 7.24299419790424720247431433939, 8.327430348856798605825583514736, 8.960934705831390552652157169407, 10.06537964926476708638967633352
