| L(s) = 1 | + (−0.594 + 0.660i)2-s + (0.138 + 1.31i)3-s + (0.126 + 1.20i)4-s + (1.85 − 1.24i)5-s + (−0.951 − 0.691i)6-s + (1.95 + 1.78i)7-s + (−2.30 − 1.67i)8-s + (1.22 − 0.259i)9-s + (−0.283 + 1.96i)10-s + (−3.42 − 0.727i)11-s + (−1.56 + 0.333i)12-s + (−1.01 + 3.13i)13-s + (−2.33 + 0.226i)14-s + (1.89 + 2.27i)15-s + (0.108 − 0.0231i)16-s + (−1.37 + 0.614i)17-s + ⋯ |
| L(s) = 1 | + (−0.420 + 0.466i)2-s + (0.0798 + 0.760i)3-s + (0.0632 + 0.602i)4-s + (0.831 − 0.556i)5-s + (−0.388 − 0.282i)6-s + (0.737 + 0.675i)7-s + (−0.815 − 0.592i)8-s + (0.406 − 0.0864i)9-s + (−0.0897 + 0.621i)10-s + (−1.03 − 0.219i)11-s + (−0.452 + 0.0962i)12-s + (−0.282 + 0.868i)13-s + (−0.625 + 0.0604i)14-s + (0.489 + 0.587i)15-s + (0.0271 − 0.00577i)16-s + (−0.334 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.735852 + 0.832780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735852 + 0.832780i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.85 + 1.24i)T \) |
| 7 | \( 1 + (-1.95 - 1.78i)T \) |
| good | 2 | \( 1 + (0.594 - 0.660i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.138 - 1.31i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (3.42 + 0.727i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.01 - 3.13i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.37 - 0.614i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.525 + 5.00i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.49 - 1.65i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-7.71 + 5.60i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.43 - 2.42i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.01 + 0.854i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-3.20 + 9.87i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 0.679T + 43T^{2} \) |
| 47 | \( 1 + (2.81 + 1.25i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-0.0147 - 0.140i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (0.679 + 0.754i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 1.62i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-11.2 + 5.01i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (6.09 - 4.42i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (10.1 + 2.15i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-8.72 - 3.88i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.18 - 3.04i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (7.43 - 8.25i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (13.0 - 9.46i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.95434156049391718108396950186, −12.05887662931111517178970981222, −10.86360184342423293061342938911, −9.618595065618449856872701811311, −8.997772248131926717789021716806, −8.096297128890184726047037742153, −6.79851796949847414441000483141, −5.38400066196773852312703257084, −4.34933988899254074884014196109, −2.45014642328837403113005121481,
1.41229894308019217060037994135, 2.61026251277442535472273662657, 4.96894993287397317316646832665, 6.14041966375721133261909471205, 7.33276697509832210522526991055, 8.258731480407546596071993840643, 9.915267919847170754857096843329, 10.32692768996207072775533948637, 11.16730844767619565818383828882, 12.52747443384373206173704907990
