L(s) = 1 | + 3·3-s − 2·5-s + 7-s + 6·9-s + 5·11-s − 2·13-s − 6·15-s + 3·21-s − 2·23-s − 25-s + 9·27-s + 6·29-s + 4·31-s + 15·33-s − 2·35-s − 37-s − 6·39-s − 9·41-s − 2·43-s − 12·45-s + 9·47-s − 6·49-s + 53-s − 10·55-s − 8·59-s − 8·61-s + 6·63-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.894·5-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.554·13-s − 1.54·15-s + 0.654·21-s − 0.417·23-s − 1/5·25-s + 1.73·27-s + 1.11·29-s + 0.718·31-s + 2.61·33-s − 0.338·35-s − 0.164·37-s − 0.960·39-s − 1.40·41-s − 0.304·43-s − 1.78·45-s + 1.31·47-s − 6/7·49-s + 0.137·53-s − 1.34·55-s − 1.04·59-s − 1.02·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.451389381\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.451389381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−10.47846947528504304597033133534, −9.546549874329867499291034737316, −8.818114582649404765195857121752, −8.123271734694736657762483607294, −7.43472452798916480523381832837, −6.49493753091628743991483348654, −4.60961859015207352761190435857, −3.89751747398550177685794998790, −2.96144840295301813739994633739, −1.60813782925681461048810108749,
1.60813782925681461048810108749, 2.96144840295301813739994633739, 3.89751747398550177685794998790, 4.60961859015207352761190435857, 6.49493753091628743991483348654, 7.43472452798916480523381832837, 8.123271734694736657762483607294, 8.818114582649404765195857121752, 9.546549874329867499291034737316, 10.47846947528504304597033133534