| L(s) = 1 | + (−0.291 + 0.265i)2-s + (2.53 + 1.56i)3-s + (−0.170 + 1.83i)4-s + (0.715 + 0.652i)5-s + (−1.15 + 0.215i)6-s + (−0.273 − 0.961i)7-s + (−0.912 − 1.20i)8-s + (2.62 + 5.26i)9-s − 0.381·10-s + (−0.479 + 5.17i)11-s + (−3.31 + 4.38i)12-s + (−1.28 − 4.50i)13-s + (0.334 + 0.207i)14-s + (0.789 + 2.77i)15-s + (−3.04 − 0.568i)16-s + (−2.41 + 3.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.205 + 0.187i)2-s + (1.46 + 0.905i)3-s + (−0.0851 + 0.918i)4-s + (0.320 + 0.291i)5-s + (−0.471 + 0.0880i)6-s + (−0.103 − 0.363i)7-s + (−0.322 − 0.427i)8-s + (0.874 + 1.75i)9-s − 0.120·10-s + (−0.144 + 1.55i)11-s + (−0.956 + 1.26i)12-s + (−0.355 − 1.24i)13-s + (0.0895 + 0.0554i)14-s + (0.203 + 0.716i)15-s + (−0.760 − 0.142i)16-s + (−0.585 + 0.774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.809053 + 2.03574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809053 + 2.03574i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.273 + 0.961i)T \) |
| 137 | \( 1 + (11.4 + 2.30i)T \) |
| good | 2 | \( 1 + (0.291 - 0.265i)T + (0.184 - 1.99i)T^{2} \) |
| 3 | \( 1 + (-2.53 - 1.56i)T + (1.33 + 2.68i)T^{2} \) |
| 5 | \( 1 + (-0.715 - 0.652i)T + (0.461 + 4.97i)T^{2} \) |
| 11 | \( 1 + (0.479 - 5.17i)T + (-10.8 - 2.02i)T^{2} \) |
| 13 | \( 1 + (1.28 + 4.50i)T + (-11.0 + 6.84i)T^{2} \) |
| 17 | \( 1 + (2.41 - 3.19i)T + (-4.65 - 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.79 - 0.695i)T + (14.0 + 12.8i)T^{2} \) |
| 23 | \( 1 + (6.81 + 1.27i)T + (21.4 + 8.30i)T^{2} \) |
| 29 | \( 1 + (-8.05 - 1.50i)T + (27.0 + 10.4i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 0.528i)T + (22.9 - 20.8i)T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 + (-5.18 - 2.00i)T + (31.7 + 28.9i)T^{2} \) |
| 47 | \( 1 + (-4.12 - 8.29i)T + (-28.3 + 37.5i)T^{2} \) |
| 53 | \( 1 + (-11.7 - 4.57i)T + (39.1 + 35.7i)T^{2} \) |
| 59 | \( 1 + (4.31 + 8.65i)T + (-35.5 + 47.0i)T^{2} \) |
| 61 | \( 1 + (4.11 - 8.25i)T + (-36.7 - 48.6i)T^{2} \) |
| 67 | \( 1 + (2.67 + 9.39i)T + (-56.9 + 35.2i)T^{2} \) |
| 71 | \( 1 + (1.32 + 14.3i)T + (-69.7 + 13.0i)T^{2} \) |
| 73 | \( 1 + (2.08 - 7.31i)T + (-62.0 - 38.4i)T^{2} \) |
| 79 | \( 1 + (8.34 - 5.16i)T + (35.2 - 70.7i)T^{2} \) |
| 83 | \( 1 + (-0.842 - 1.11i)T + (-22.7 + 79.8i)T^{2} \) |
| 89 | \( 1 + (-2.69 - 2.45i)T + (8.21 + 88.6i)T^{2} \) |
| 97 | \( 1 + (-0.830 + 8.95i)T + (-95.3 - 17.8i)T^{2} \) |
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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
−10.09575225705948252768291511680, −9.552501395088615638345159138598, −8.624852902999970921727170941837, −7.85816697400151429436540245992, −7.48690812954533458007077824805, −6.22983574890233253930416796029, −4.52560194898777442520720760244, −4.15160105758016673107535778098, −2.93342456162096513967318662997, −2.33738888783146872885065954584,
0.925453465323881296010512177820, 2.10892770499313470248784956898, 2.80926566273982817643036217368, 4.18948827986689157965355377046, 5.55887967306097112458356402852, 6.36369807682191753849467878114, 7.25200594465543128663499259885, 8.331462828393155736942864906720, 8.961720445428180138293042406876, 9.352994186024838658320924137773
