L(s) = 1 | − 2.72·2-s + 3-s + 5.40·4-s − 2.75·5-s − 2.72·6-s + 0.631·7-s − 9.27·8-s + 9-s + 7.51·10-s + 0.617·11-s + 5.40·12-s + 2.46·13-s − 1.72·14-s − 2.75·15-s + 14.4·16-s − 17-s − 2.72·18-s + 2.03·19-s − 14.9·20-s + 0.631·21-s − 1.68·22-s + 3.83·23-s − 9.27·24-s + 2.61·25-s − 6.70·26-s + 27-s + 3.41·28-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.70·4-s − 1.23·5-s − 1.11·6-s + 0.238·7-s − 3.27·8-s + 0.333·9-s + 2.37·10-s + 0.186·11-s + 1.56·12-s + 0.683·13-s − 0.459·14-s − 0.712·15-s + 3.60·16-s − 0.242·17-s − 0.641·18-s + 0.466·19-s − 3.33·20-s + 0.137·21-s − 0.358·22-s + 0.800·23-s − 1.89·24-s + 0.523·25-s − 1.31·26-s + 0.192·27-s + 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 - 0.631T + 7T^{2} \) |
| 11 | \( 1 - 0.617T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 - 1.97T + 61T^{2} \) |
| 67 | \( 1 + 5.16T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 - 6.89T + 97T^{2} \) |
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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
−8.088389257059713257393478327549, −7.66535638517349313081182535435, −7.08396587857958714668169780692, −6.36400015093689458044638068308, −5.19967790644752519191910586998, −3.78958253695917404423430631996, −3.27258831730530033545807698198, −2.10502500111265867576986125240, −1.20340289813640878301603129458, 0,
1.20340289813640878301603129458, 2.10502500111265867576986125240, 3.27258831730530033545807698198, 3.78958253695917404423430631996, 5.19967790644752519191910586998, 6.36400015093689458044638068308, 7.08396587857958714668169780692, 7.66535638517349313081182535435, 8.088389257059713257393478327549