| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s + 4·11-s − 12-s − 2·13-s − 4·14-s + 16-s + 2·17-s + 18-s + 4·21-s + 4·22-s − 24-s − 2·26-s − 27-s − 4·28-s − 6·29-s + 32-s − 4·33-s + 2·34-s + 36-s + 37-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.872·21-s + 0.852·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + 0.164·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.46517272202628311515668415288, −6.70710953001223829275720467596, −6.49598433597014765643700749556, −5.62087280221581461301517418841, −5.00442386516073654060856947404, −3.88111969711942643405789311836, −3.56450351882555114378909755216, −2.53446603802029530182391324523, −1.36049628108465181935555829530, 0,
1.36049628108465181935555829530, 2.53446603802029530182391324523, 3.56450351882555114378909755216, 3.88111969711942643405789311836, 5.00442386516073654060856947404, 5.62087280221581461301517418841, 6.49598433597014765643700749556, 6.70710953001223829275720467596, 7.46517272202628311515668415288
