| L(s) = 1 | + 0.458·2-s − 0.533·3-s − 1.79·4-s + 5-s − 0.244·6-s + 7-s − 1.73·8-s − 2.71·9-s + 0.458·10-s − 1.32·11-s + 0.954·12-s + 3.70·13-s + 0.458·14-s − 0.533·15-s + 2.78·16-s + 0.727·17-s − 1.24·18-s + 3.95·19-s − 1.79·20-s − 0.533·21-s − 0.607·22-s − 2.95·23-s + 0.926·24-s + 25-s + 1.69·26-s + 3.04·27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | + 0.324·2-s − 0.307·3-s − 0.895·4-s + 0.447·5-s − 0.0997·6-s + 0.377·7-s − 0.614·8-s − 0.905·9-s + 0.144·10-s − 0.399·11-s + 0.275·12-s + 1.02·13-s + 0.122·14-s − 0.137·15-s + 0.696·16-s + 0.176·17-s − 0.293·18-s + 0.906·19-s − 0.400·20-s − 0.116·21-s − 0.129·22-s − 0.616·23-s + 0.189·24-s + 0.200·25-s + 0.333·26-s + 0.586·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 0.458T + 2T^{2} \) |
| 3 | \( 1 + 0.533T + 3T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 0.727T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 0.892T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 + 0.183T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 5.22T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 - 6.59T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.59819415942584883094779700588, −6.58808981158956893522239888030, −5.84299456672412102090181725091, −5.35048791858688405794058521612, −4.95961152625149154096508647713, −3.76058441984464588488048702534, −3.39749292130831399091605593681, −2.25667083070844271705859068389, −1.16052320166762169603541039391, 0,
1.16052320166762169603541039391, 2.25667083070844271705859068389, 3.39749292130831399091605593681, 3.76058441984464588488048702534, 4.95961152625149154096508647713, 5.35048791858688405794058521612, 5.84299456672412102090181725091, 6.58808981158956893522239888030, 7.59819415942584883094779700588
