| L(s) = 1 | + (1.30 + 0.541i)2-s + (−0.391 − 1.96i)3-s + (1.41 + 1.41i)4-s + (1.85 + 1.24i)5-s + (0.553 − 2.78i)6-s + (3.61 − 2.41i)7-s + (1.08 + 2.61i)8-s + (4.59 − 1.90i)9-s + (1.75 + 2.62i)10-s + (−2.11 − 0.421i)11-s + (2.22 − 3.33i)12-s + (1.60 − 1.60i)13-s + (6.02 − 1.19i)14-s + (1.71 − 4.14i)15-s + 4i·16-s + (16.3 − 4.63i)17-s + ⋯ |
| L(s) = 1 | + (0.653 + 0.270i)2-s + (−0.130 − 0.655i)3-s + (0.353 + 0.353i)4-s + (0.371 + 0.248i)5-s + (0.0922 − 0.463i)6-s + (0.516 − 0.344i)7-s + (0.135 + 0.326i)8-s + (0.510 − 0.211i)9-s + (0.175 + 0.262i)10-s + (−0.192 − 0.0382i)11-s + (0.185 − 0.278i)12-s + (0.123 − 0.123i)13-s + (0.430 − 0.0856i)14-s + (0.114 − 0.276i)15-s + 0.250i·16-s + (0.962 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.30496 - 0.200433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.30496 - 0.200433i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 0.541i)T \) |
| 5 | \( 1 + (-1.85 - 1.24i)T \) |
| 17 | \( 1 + (-16.3 + 4.63i)T \) |
| good | 3 | \( 1 + (0.391 + 1.96i)T + (-8.31 + 3.44i)T^{2} \) |
| 7 | \( 1 + (-3.61 + 2.41i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (2.11 + 0.421i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 1.60i)T - 169iT^{2} \) |
| 19 | \( 1 + (-0.919 - 0.381i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (2.62 - 13.2i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-0.955 + 1.43i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (54.1 - 10.7i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-0.112 - 0.567i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (45.4 - 30.3i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (17.5 - 7.26i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (32.0 - 32.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-36.6 - 15.1i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (7.00 + 16.8i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (36.9 + 55.3i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 5.01iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (22.9 + 115. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-34.0 - 22.7i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (13.0 + 2.59i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-2.98 + 7.20i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-75.1 - 75.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (12.6 - 18.9i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−12.67456438837406125545261282470, −11.75344134480626392734032632987, −10.70651589430594255670973531949, −9.551503001398751577886964038438, −7.945476559425833399957014965028, −7.21062835107344375520234627585, −6.12521588262694861535160386766, −4.97424646049642256942449971201, −3.44351844800333005376351516557, −1.61011098537962869584114295163,
1.85149804343729159990989361936, 3.67583404517344295999776520083, 4.90721088955033325359043373910, 5.69102341305848297590049718402, 7.22360319738308664502414895533, 8.614531654473135161052170831380, 9.849458759876286117991302641336, 10.56381757915615373241770417216, 11.61377089105328066535349255816, 12.60190722056608651490080422277
