| L(s) = 1 | + (−1.19 − 0.769i)2-s + (−0.989 + 0.142i)3-s + (0.0103 + 0.0227i)4-s + (1.85 + 0.543i)5-s + (1.29 + 0.591i)6-s + (1.99 + 1.74i)7-s + (−0.399 + 2.78i)8-s + (0.959 − 0.281i)9-s + (−1.79 − 2.07i)10-s + (−1.95 − 3.04i)11-s + (−0.0135 − 0.0210i)12-s + (−2.98 + 2.58i)13-s + (−1.04 − 3.61i)14-s + (−1.90 − 0.274i)15-s + (2.65 − 3.06i)16-s + (−2.49 + 5.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.846 − 0.544i)2-s + (−0.571 + 0.0821i)3-s + (0.00519 + 0.0113i)4-s + (0.827 + 0.242i)5-s + (0.528 + 0.241i)6-s + (0.752 + 0.658i)7-s + (−0.141 + 0.983i)8-s + (0.319 − 0.0939i)9-s + (−0.568 − 0.655i)10-s + (−0.590 − 0.918i)11-s + (−0.00390 − 0.00607i)12-s + (−0.827 + 0.717i)13-s + (−0.278 − 0.966i)14-s + (−0.492 − 0.0708i)15-s + (0.662 − 0.765i)16-s + (−0.604 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.627591 + 0.267976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627591 + 0.267976i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (-1.99 - 1.74i)T \) |
| 23 | \( 1 + (4.57 - 1.44i)T \) |
| good | 2 | \( 1 + (1.19 + 0.769i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.85 - 0.543i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (1.95 + 3.04i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.98 - 2.58i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.49 - 5.45i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.11 - 4.62i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.273 - 0.599i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-9.09 - 1.30i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 5.10i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.999 + 3.40i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-11.5 + 1.65i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 8.64iT - 47T^{2} \) |
| 53 | \( 1 + (4.59 + 3.98i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (2.75 - 2.39i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.18 - 8.21i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (2.51 - 3.92i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.52 - 1.62i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.80 - 0.824i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.73 + 7.56i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (1.39 - 0.410i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.896 - 6.23i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (9.51 + 2.79i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.89562890610336561356581849366, −10.23125975598932996595371129238, −9.555930228496214535491639602701, −8.546775751548501158098739026624, −7.83184044837492615571942096348, −6.08389644891385890295104711670, −5.72613133009925317107145742900, −4.50820823828511540543555900700, −2.52089772336836827638956057170, −1.54963352800275006838340462511,
0.60451553732400795309915391459, 2.40995241930767455515153740351, 4.46664085172018075092040574964, 5.17997858917008233758112567310, 6.48511753888550226911470679355, 7.45528804773479990301024342097, 7.85816882726793065619476446836, 9.256757856760008715584879431410, 9.814874537993523146944510729455, 10.56525068845842240011851461498
