L(s) = 1 | + (−1.27 − 2.20i)2-s + (1.86 − 1.07i)3-s + (−2.24 + 3.88i)4-s + (2.08 − 0.817i)5-s + (−4.74 − 2.74i)6-s + (1.46 − 2.54i)7-s + 6.31·8-s + (0.817 − 1.41i)9-s + (−4.45 − 3.54i)10-s + (0.550 − 0.317i)11-s + 9.64i·12-s − 7.48·14-s + (3.00 − 3.76i)15-s + (−3.55 − 6.16i)16-s + (−1.05 − 0.611i)17-s − 4.16·18-s + ⋯ |
L(s) = 1 | + (−0.900 − 1.55i)2-s + (1.07 − 0.621i)3-s + (−1.12 + 1.94i)4-s + (0.930 − 0.365i)5-s + (−1.93 − 1.11i)6-s + (0.555 − 0.961i)7-s + 2.23·8-s + (0.272 − 0.472i)9-s + (−1.40 − 1.12i)10-s + (0.165 − 0.0957i)11-s + 2.78i·12-s − 1.99·14-s + (0.774 − 0.972i)15-s + (−0.889 − 1.54i)16-s + (−0.257 − 0.148i)17-s − 0.981·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0964455 - 1.62027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0964455 - 1.62027i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.08 + 0.817i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.27 + 2.20i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.86 + 1.07i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.46 + 2.54i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.550 + 0.317i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.05 + 0.611i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 0.682i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 1.07i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.96iT - 31T^{2} \) |
| 37 | \( 1 + (0.611 + 1.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.62 - 4.98i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.18 - 0.683i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 0.642iT - 53T^{2} \) |
| 59 | \( 1 + (6.57 + 3.79i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.01 - 6.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.28 + 1.31i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + (10.8 - 6.27i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.39 - 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.682808262320001400986215697326, −9.161882805421908406107693184625, −8.276687412203053325974390756273, −7.80604691304523092538120160377, −6.68292226203745463232742311349, −4.96058845163213066750923055855, −3.81031738762480984476688916405, −2.75330443293106619955581701817, −1.87058855597632314394517515940, −1.04112708327430678318108676608,
1.78966912075947022681747115905, 3.10627714468054138843260620104, 4.77647901873233651983161471953, 5.52209054427145397878669145095, 6.43869870703973293683063394500, 7.25225989118961750063737163367, 8.477352309175134299537102016706, 8.637487764522161501491689892146, 9.485674862363593130255674552649, 9.973000165871202605964584901005