| L(s) = 1 | + (0.826 − 0.563i)2-s + (−2.19 + 2.03i)3-s + (0.365 − 0.930i)4-s + (3.17 + 0.980i)5-s + (−0.665 + 2.91i)6-s + (−0.0867 + 2.64i)7-s + (−0.222 − 0.974i)8-s + (0.444 − 5.92i)9-s + (3.17 − 0.980i)10-s + (−0.138 − 1.84i)11-s + (1.09 + 2.78i)12-s + (−2.68 − 1.29i)13-s + (1.41 + 2.23i)14-s + (−8.96 + 4.31i)15-s + (−0.733 − 0.680i)16-s + (−1.81 − 0.273i)17-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.398i)2-s + (−1.26 + 1.17i)3-s + (0.182 − 0.465i)4-s + (1.42 + 0.438i)5-s + (−0.271 + 1.19i)6-s + (−0.0327 + 0.999i)7-s + (−0.0786 − 0.344i)8-s + (0.148 − 1.97i)9-s + (1.00 − 0.310i)10-s + (−0.0417 − 0.556i)11-s + (0.315 + 0.803i)12-s + (−0.745 − 0.359i)13-s + (0.378 + 0.596i)14-s + (−2.31 + 1.11i)15-s + (−0.183 − 0.170i)16-s + (−0.439 − 0.0662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05302 + 0.323980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05302 + 0.323980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 + (0.0867 - 2.64i)T \) |
| good | 3 | \( 1 + (2.19 - 2.03i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-3.17 - 0.980i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.138 + 1.84i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (2.68 + 1.29i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.81 + 0.273i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.91 + 5.04i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.233 - 0.0352i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 1.26i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.552 + 0.956i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.957 - 2.43i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.45 - 10.7i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.82 + 12.3i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (5.91 - 4.03i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (1.52 - 3.89i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (9.88 - 3.04i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (3.53 + 8.99i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.11 - 5.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.61 - 7.04i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.05 - 0.721i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.35 - 1.61i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.736 + 9.82i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−14.03516966014020909862746573498, −12.85342168514274890347874666953, −11.68379509513378166612232122615, −10.92045463849898768347471768918, −9.908644092877110231874059385952, −9.267539529256527148356343582568, −6.45262961444539907117720380500, −5.63956412108679641975847644212, −4.85914861846519527907010337738, −2.79231751563585162382948137132,
1.74326807488123590142448256853, 4.78630828443294338822227013778, 5.81820231001389479001763464030, 6.74487674333323857200225653544, 7.66989089975609217303481675395, 9.683258852235532876187993076791, 10.81972303679682595196268578153, 12.14916326736909278783771519659, 12.82408136860041931361879258381, 13.64040135226852924054658913874
