| L(s) = 1 | + (1.74 + 1.37i)2-s + (0.814 + 0.580i)3-s + (0.691 + 2.84i)4-s + (0.900 − 0.173i)5-s + (0.625 + 2.13i)6-s + (−2.36 + 1.19i)7-s + (−0.859 + 1.88i)8-s + (0.327 + 0.945i)9-s + (1.80 + 0.933i)10-s + (1.96 + 2.49i)11-s + (−1.08 + 2.72i)12-s + (−1.26 − 1.96i)13-s + (−5.76 − 1.15i)14-s + (0.834 + 0.380i)15-s + (1.12 − 0.581i)16-s + (0.328 + 0.312i)17-s + ⋯ |
| L(s) = 1 | + (1.23 + 0.970i)2-s + (0.470 + 0.334i)3-s + (0.345 + 1.42i)4-s + (0.402 − 0.0776i)5-s + (0.255 + 0.869i)6-s + (−0.892 + 0.451i)7-s + (−0.303 + 0.665i)8-s + (0.109 + 0.315i)9-s + (0.572 + 0.295i)10-s + (0.592 + 0.752i)11-s + (−0.314 + 0.785i)12-s + (−0.350 − 0.545i)13-s + (−1.53 − 0.309i)14-s + (0.215 + 0.0983i)15-s + (0.281 − 0.145i)16-s + (0.0795 + 0.0758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89109 + 2.40986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89109 + 2.40986i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.814 - 0.580i)T \) |
| 7 | \( 1 + (2.36 - 1.19i)T \) |
| 23 | \( 1 + (-1.31 + 4.61i)T \) |
| good | 2 | \( 1 + (-1.74 - 1.37i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.900 + 0.173i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.96 - 2.49i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (1.26 + 1.96i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.328 - 0.312i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.682 - 0.651i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 0.318i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.611 + 6.39i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-8.69 + 3.00i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-4.93 + 4.27i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (5.82 - 2.65i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (10.0 + 5.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.57 + 0.313i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (0.163 - 0.317i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (1.64 + 2.30i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-4.39 - 10.9i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (1.01 - 7.07i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.33 + 1.53i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (2.20 - 0.104i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (2.33 - 2.69i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (17.1 + 1.64i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 1.50i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.52743137152605159442875259060, −9.987842343754822330826641259351, −9.575729374452082151185421746563, −8.335425447588853104092472139037, −7.32551552338871763202364605079, −6.40531552757876770173137089467, −5.66273144309600192789117612369, −4.59393558790771599215540819037, −3.67130034806619943247579747927, −2.48584213013371295465991190753,
1.48839456483869305409066289675, 2.86421180823605322427353760901, 3.58700152487592791843234456233, 4.66946597156515593458619924548, 5.99076743150833150296820301637, 6.67053583530830271341152023781, 7.977882774811449042771519569130, 9.316407582858001193431571387799, 9.918853716945182710917131525496, 11.02650295635655404289464904932
