L(s) = 1 | + (1.22 − 0.707i)2-s + (1.22 + 2.12i)5-s + (2.5 − 0.866i)7-s + 2.82i·8-s + (3 + 1.73i)10-s + (−4.89 − 2.82i)11-s − 3.46i·13-s + (2.44 − 2.82i)14-s + (2.00 + 3.46i)16-s + (1.22 − 2.12i)17-s + (1.5 − 0.866i)19-s − 8·22-s + (−6.12 + 3.53i)23-s + (−0.499 + 0.866i)25-s + (−2.44 − 4.24i)26-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)2-s + (0.547 + 0.948i)5-s + (0.944 − 0.327i)7-s + 0.999i·8-s + (0.948 + 0.547i)10-s + (−1.47 − 0.852i)11-s − 0.960i·13-s + (0.654 − 0.755i)14-s + (0.500 + 0.866i)16-s + (0.297 − 0.514i)17-s + (0.344 − 0.198i)19-s − 1.70·22-s + (−1.27 + 0.737i)23-s + (−0.0999 + 0.173i)25-s + (−0.480 − 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84164 - 0.0583769i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84164 - 0.0583769i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.22 + 0.707i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.89 + 2.82i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.22 + 0.707i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.34 + 12.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (-10.5 - 6.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + (1.22 + 2.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−12.72661677005300829742858186927, −11.46164323713634425751724908063, −10.87698903361428098748662922302, −10.00645608608081351561977265559, −8.238439029753145795952760410200, −7.59746714390585447706212114441, −5.80396961269920167739436713063, −5.05064475941037451959452346539, −3.43660598060250442539838764466, −2.43187754419772588526089620704,
1.90748579708039254720588099870, 4.28653418143323724872235954554, 5.11314226612979465325326970094, 5.83485878016172540859065191283, 7.33341581612845252273168719148, 8.463681845534553039326776225981, 9.601338032656928756492484740107, 10.51806291731875016993377065202, 12.04998779011347332269161822310, 12.73760991071333305986867256581