| L(s) = 1 | + (−0.983 + 2.07i)2-s + (1.66 + 1.53i)3-s + (−2.06 − 2.52i)4-s + (−1.83 + 1.27i)5-s + (−4.80 + 1.93i)6-s + (−1.69 − 0.492i)7-s + (2.80 − 0.691i)8-s + (0.169 + 2.09i)9-s + (−0.823 − 5.06i)10-s + (0.718 + 0.844i)11-s + (0.444 − 7.35i)12-s + (−0.898 + 3.49i)13-s + (2.68 − 3.03i)14-s + (−5.00 − 0.708i)15-s + (−0.0219 + 0.107i)16-s + (−0.881 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.695 + 1.46i)2-s + (0.958 + 0.884i)3-s + (−1.03 − 1.26i)4-s + (−0.822 + 0.568i)5-s + (−1.96 + 0.789i)6-s + (−0.642 − 0.185i)7-s + (0.992 − 0.244i)8-s + (0.0564 + 0.698i)9-s + (−0.260 − 1.60i)10-s + (0.216 + 0.254i)11-s + (0.128 − 2.12i)12-s + (−0.249 + 0.968i)13-s + (0.718 − 0.811i)14-s + (−1.29 − 0.182i)15-s + (−0.00547 + 0.0268i)16-s + (−0.213 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.303373 - 0.163316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.303373 - 0.163316i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.83 - 1.27i)T \) |
| 13 | \( 1 + (0.898 - 3.49i)T \) |
| good | 2 | \( 1 + (0.983 - 2.07i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-1.66 - 1.53i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (1.69 + 0.492i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-0.718 - 0.844i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (0.881 + 1.60i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (6.95 - 1.86i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.286 + 1.06i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.29 + 0.613i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.123 - 0.0966i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (0.579 + 0.770i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (0.611 + 0.564i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.235 - 1.66i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (-11.6 + 4.41i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-1.72 + 2.85i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (4.37 + 2.88i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (8.73 - 9.09i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (2.01 - 2.47i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (2.87 - 12.7i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (6.86 - 9.94i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (12.5 - 4.77i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (4.35 + 8.29i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (12.1 + 3.26i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.53 + 1.51i)T + (77.5 - 58.2i)T^{2} \) |
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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
−10.31915321123429665565029841427, −9.841101742793263424293631478715, −8.800027864082870163802865941175, −8.620341620276390219843772477158, −7.44679437929632705920168964100, −6.88232022981500899299226658786, −6.07169657676345423025471827777, −4.51761103867978609102765214442, −3.92285319279853216907546365735, −2.64808264832709883894335877754,
0.18632086863660966389996188878, 1.53813153650329151742530571598, 2.67871248079086870983991189820, 3.38434550092471325666305328304, 4.44450787801710741311847533319, 6.13457022048061814529577475963, 7.39565982190211886179020500314, 8.128918623101136404068064631445, 8.805559346547087475574954969636, 9.241053365992292821642487308339
