| L(s) = 1 | + (0.337 + 0.150i)2-s + (−1.17 − 1.27i)3-s + (−1.24 − 1.38i)4-s + (−0.966 − 2.17i)5-s + (−0.204 − 0.607i)6-s + (0.327 + 1.54i)7-s + (−0.441 − 1.35i)8-s + (−0.251 + 2.98i)9-s − 0.879i·10-s + (2.12 − 2.54i)11-s + (−0.303 + 3.21i)12-s + (6.09 − 0.640i)13-s + (−0.121 + 0.570i)14-s + (−1.63 + 3.77i)15-s + (−0.334 + 3.17i)16-s + (−0.887 − 0.644i)17-s + ⋯ |
| L(s) = 1 | + (0.238 + 0.106i)2-s + (−0.676 − 0.736i)3-s + (−0.623 − 0.692i)4-s + (−0.432 − 0.970i)5-s + (−0.0834 − 0.247i)6-s + (0.123 + 0.583i)7-s + (−0.156 − 0.480i)8-s + (−0.0837 + 0.996i)9-s − 0.278i·10-s + (0.641 − 0.766i)11-s + (−0.0876 + 0.927i)12-s + (1.69 − 0.177i)13-s + (−0.0324 + 0.152i)14-s + (−0.422 + 0.975i)15-s + (−0.0835 + 0.794i)16-s + (−0.215 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.517961 - 0.582962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.517961 - 0.582962i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.17 + 1.27i)T \) |
| 11 | \( 1 + (-2.12 + 2.54i)T \) |
| good | 2 | \( 1 + (-0.337 - 0.150i)T + (1.33 + 1.48i)T^{2} \) |
| 5 | \( 1 + (0.966 + 2.17i)T + (-3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (-0.327 - 1.54i)T + (-6.39 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-6.09 + 0.640i)T + (12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.887 + 0.644i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.534 - 0.173i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.81 + 2.78i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.726 - 0.154i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.661 - 6.29i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.32 + 4.07i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-11.8 - 2.52i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (7.35 - 4.24i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.571 - 0.514i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-2.66 - 3.66i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.00834 - 0.00751i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-5.69 - 0.599i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (0.738 - 1.27i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.89 - 6.74i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (14.9 + 4.87i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.29 + 2.90i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.08 - 10.3i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 - 9.01iT - 89T^{2} \) |
| 97 | \( 1 + (-5.06 - 2.25i)T + (64.9 + 72.0i)T^{2} \) |
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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
−13.47222893115412443507728464362, −12.67758288613354770647872751353, −11.67469135394072231461621982402, −10.65288749674594274100069174356, −8.927779274166070152266780760674, −8.311500499719828813049117085737, −6.33859655730666033429293909598, −5.58882733969301047995689374785, −4.24415067638477347470861048544, −1.08062158354187521553300891840,
3.63568158465370532249933751911, 4.27179080895124452638711094156, 6.09823362253657951982840253952, 7.39486674353056747037574936619, 8.829093914879412241297748860226, 10.06918847778153428975031590196, 11.19034447772516286599085909195, 11.79592830860987491174825676963, 13.15508044172678401450434623712, 14.21106450141561955960195875168
