| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s − 6·17-s + 18-s − 19-s + 20-s + 4·21-s + 24-s + 25-s − 2·26-s + 27-s + 4·28-s + 6·29-s + 30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.39911518506784, −13.92953476654777, −13.44468983444611, −13.16878937087610, −12.35641511791182, −12.02558065878227, −11.41806435971751, −10.86448847430899, −10.60106739808385, −9.848582635473767, −9.306913206539952, −8.622891339303429, −8.313468500227445, −7.740317728297312, −7.092984798669490, −6.665784699612324, −6.023500979974420, −5.277720843419152, −4.760035400260440, −4.497747912733129, −3.822145920480578, −2.895775009256596, −2.499484076572193, −1.681104388325211, −1.472311606969330, 0,
1.472311606969330, 1.681104388325211, 2.499484076572193, 2.895775009256596, 3.822145920480578, 4.497747912733129, 4.760035400260440, 5.277720843419152, 6.023500979974420, 6.665784699612324, 7.092984798669490, 7.740317728297312, 8.313468500227445, 8.622891339303429, 9.306913206539952, 9.848582635473767, 10.60106739808385, 10.86448847430899, 11.41806435971751, 12.02558065878227, 12.35641511791182, 13.16878937087610, 13.44468983444611, 13.92953476654777, 14.39911518506784
