| L(s) = 1 | + 0.879·2-s + 3-s − 1.22·4-s − 2·5-s + 0.879·6-s − 2.87·7-s − 2.83·8-s + 9-s − 1.75·10-s − 6.41·11-s − 1.22·12-s + 0.694·13-s − 2.53·14-s − 2·15-s − 0.0418·16-s + 1.06·17-s + 0.879·18-s + 0.532·19-s + 2.45·20-s − 2.87·21-s − 5.63·22-s + 7.00·23-s − 2.83·24-s − 25-s + 0.610·26-s + 27-s + 3.53·28-s + ⋯ |
| L(s) = 1 | + 0.621·2-s + 0.577·3-s − 0.613·4-s − 0.894·5-s + 0.359·6-s − 1.08·7-s − 1.00·8-s + 0.333·9-s − 0.556·10-s − 1.93·11-s − 0.354·12-s + 0.192·13-s − 0.676·14-s − 0.516·15-s − 0.0104·16-s + 0.258·17-s + 0.207·18-s + 0.122·19-s + 0.548·20-s − 0.628·21-s − 1.20·22-s + 1.46·23-s − 0.579·24-s − 0.200·25-s + 0.119·26-s + 0.192·27-s + 0.667·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 13 | \( 1 - 0.694T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 0.532T + 19T^{2} \) |
| 23 | \( 1 - 7.00T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 + 0.980T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 1.06T + 79T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 - 1.44T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−10.55330371639105575670813474098, −9.736417693501252336454827331524, −8.761812575142785060153357315940, −7.964885555514642885195162624732, −7.02677352044105490682001533965, −5.64950610658375431370518999612, −4.72933430146080380396574834337, −3.50435671725555125682029140222, −2.92160724940395528476917274602, 0,
2.92160724940395528476917274602, 3.50435671725555125682029140222, 4.72933430146080380396574834337, 5.64950610658375431370518999612, 7.02677352044105490682001533965, 7.964885555514642885195162624732, 8.761812575142785060153357315940, 9.736417693501252336454827331524, 10.55330371639105575670813474098
