| L(s) = 1 | − 1.69·2-s − 2.67·3-s + 0.864·4-s + 3.80·5-s + 4.52·6-s − 4.13·7-s + 1.92·8-s + 4.14·9-s − 6.44·10-s − 11-s − 2.31·12-s + 0.566·13-s + 7.00·14-s − 10.1·15-s − 4.98·16-s + 17-s − 7.02·18-s − 4.77·19-s + 3.29·20-s + 11.0·21-s + 1.69·22-s − 4.27·23-s − 5.13·24-s + 9.47·25-s − 0.958·26-s − 3.07·27-s − 3.57·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 1.54·3-s + 0.432·4-s + 1.70·5-s + 1.84·6-s − 1.56·7-s + 0.679·8-s + 1.38·9-s − 2.03·10-s − 0.301·11-s − 0.667·12-s + 0.157·13-s + 1.87·14-s − 2.62·15-s − 1.24·16-s + 0.242·17-s − 1.65·18-s − 1.09·19-s + 0.735·20-s + 2.41·21-s + 0.360·22-s − 0.892·23-s − 1.04·24-s + 1.89·25-s − 0.187·26-s − 0.590·27-s − 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 13 | \( 1 - 0.566T + 13T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 0.147T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 8.98T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 - 0.350T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−7.21728052935338929269786830985, −6.70864468409366069576280751516, −6.07880242566680898062957878178, −5.79908109365885063140952010189, −4.94135718492076101586186375475, −4.06557052336510359961345580527, −2.72762325588621265852160430348, −1.87931273959868803804911630895, −0.887192678771727235347933778024, 0,
0.887192678771727235347933778024, 1.87931273959868803804911630895, 2.72762325588621265852160430348, 4.06557052336510359961345580527, 4.94135718492076101586186375475, 5.79908109365885063140952010189, 6.07880242566680898062957878178, 6.70864468409366069576280751516, 7.21728052935338929269786830985
