| L(s) = 1 | + 2.69·2-s − 3-s + 5.25·4-s + 1.04·5-s − 2.69·6-s − 7-s + 8.76·8-s + 9-s + 2.82·10-s + 0.180·11-s − 5.25·12-s − 4.89·13-s − 2.69·14-s − 1.04·15-s + 13.1·16-s + 2.82·17-s + 2.69·18-s − 0.180·19-s + 5.51·20-s + 21-s + 0.487·22-s + 23-s − 8.76·24-s − 3.89·25-s − 13.1·26-s − 27-s − 5.25·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.62·4-s + 0.469·5-s − 1.09·6-s − 0.377·7-s + 3.09·8-s + 0.333·9-s + 0.893·10-s + 0.0545·11-s − 1.51·12-s − 1.35·13-s − 0.719·14-s − 0.270·15-s + 3.27·16-s + 0.685·17-s + 0.634·18-s − 0.0415·19-s + 1.23·20-s + 0.218·21-s + 0.103·22-s + 0.208·23-s − 1.78·24-s − 0.779·25-s − 2.58·26-s − 0.192·27-s − 0.993·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.717498947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.717498947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 11 | \( 1 - 0.180T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 0.825T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 - 0.125T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 - 9.01T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.45587367010053708161694142881, −10.34874618244227510768710695402, −9.633940057648742969770375383267, −7.68643466691953921231116998944, −6.91245582849953401717626643510, −5.95983080104361308225981473247, −5.32095843718071002629300042888, −4.39685572342532568633757793783, −3.22512993107701523218861468178, −2.01093032880331381040071917599,
2.01093032880331381040071917599, 3.22512993107701523218861468178, 4.39685572342532568633757793783, 5.32095843718071002629300042888, 5.95983080104361308225981473247, 6.91245582849953401717626643510, 7.68643466691953921231116998944, 9.633940057648742969770375383267, 10.34874618244227510768710695402, 11.45587367010053708161694142881
