| L(s) = 1 | + (0.599 − 0.666i)2-s + (0.0597 + 0.568i)3-s + (0.125 + 1.19i)4-s + (−1.98 + 1.03i)5-s + (0.414 + 0.301i)6-s + (0.154 + 2.64i)7-s + (2.31 + 1.68i)8-s + (2.61 − 0.555i)9-s + (−0.498 + 1.94i)10-s + (−0.258 − 0.0549i)11-s + (−0.669 + 0.142i)12-s + (1.90 − 5.87i)13-s + (1.85 + 1.48i)14-s + (−0.707 − 1.06i)15-s + (0.171 − 0.0364i)16-s + (−4.81 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.424 − 0.471i)2-s + (0.0345 + 0.328i)3-s + (0.0625 + 0.595i)4-s + (−0.886 + 0.463i)5-s + (0.169 + 0.122i)6-s + (0.0583 + 0.998i)7-s + (0.819 + 0.595i)8-s + (0.871 − 0.185i)9-s + (−0.157 + 0.613i)10-s + (−0.0779 − 0.0165i)11-s + (−0.193 + 0.0410i)12-s + (0.529 − 1.62i)13-s + (0.494 + 0.395i)14-s + (−0.182 − 0.274i)15-s + (0.0428 − 0.00910i)16-s + (−1.16 + 0.519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27797 + 0.471771i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27797 + 0.471771i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.98 - 1.03i)T \) |
| 7 | \( 1 + (-0.154 - 2.64i)T \) |
| good | 2 | \( 1 + (-0.599 + 0.666i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.0597 - 0.568i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (0.258 + 0.0549i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.90 + 5.87i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.81 - 2.14i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.284 + 2.70i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.44 - 1.60i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.07 + 2.23i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.89 + 3.51i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 0.263i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (0.210 - 0.648i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 0.914T + 43T^{2} \) |
| 47 | \( 1 + (7.35 + 3.27i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (1.04 + 9.93i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.41 - 6.01i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.996 + 1.10i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.45 - 1.53i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-10.2 + 7.43i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.15 + 0.457i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (1.98 + 0.883i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (5.34 + 3.88i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.02 - 6.69i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (8.36 - 6.07i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.77738331848028157883987167668, −11.83462684936143344328175198226, −11.08788638015740913891220937538, −10.11264557049589150586918648432, −8.547110293442745968661735970550, −7.88354777058050788433438677034, −6.53033662851586738021600725384, −4.86191113700042797559355167291, −3.75206577874330434968283112067, −2.64988351723116384608121202753,
1.36884425694751789018301854194, 4.19723528438188066993172577636, 4.64309860334355427420668092620, 6.58478897257359651017404311707, 7.07586114768199752556548224123, 8.283841233147177997733838347928, 9.642408186608622481859828577505, 10.71531895920901953638797176785, 11.63627115535438493542582458647, 12.83568000890295481510433417871
