L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 6·11-s − 12-s + 2·13-s − 2·14-s + 15-s + 16-s + 18-s − 2·19-s − 20-s + 2·21-s − 6·22-s + 6·23-s − 24-s + 25-s + 2·26-s − 27-s − 2·28-s − 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.92297174815646, −13.89523400779452, −13.53001605955404, −13.07658294880642, −12.67518032656746, −12.31014352601159, −11.57578809541221, −11.09283290699288, −10.70848392132374, −10.20981251782560, −9.712721121297546, −8.871363076551625, −8.360807777443719, −7.644322183524463, −7.310858307586859, −6.557748596136550, −6.185175420373327, −5.515952014732357, −4.981705589037200, −4.588290747253822, −3.728786797645304, −3.195509187982656, −2.672544087921360, −1.861215554461711, −0.8008856204830296, 0,
0.8008856204830296, 1.861215554461711, 2.672544087921360, 3.195509187982656, 3.728786797645304, 4.588290747253822, 4.981705589037200, 5.515952014732357, 6.185175420373327, 6.557748596136550, 7.310858307586859, 7.644322183524463, 8.360807777443719, 8.871363076551625, 9.712721121297546, 10.20981251782560, 10.70848392132374, 11.09283290699288, 11.57578809541221, 12.31014352601159, 12.67518032656746, 13.07658294880642, 13.53001605955404, 13.89523400779452, 14.92297174815646