| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 15-s − 6·17-s − 4·19-s − 8·23-s + 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 10·37-s + 6·41-s + 4·43-s + 45-s − 7·49-s + 6·51-s + 10·53-s − 4·55-s + 4·57-s − 4·59-s + 2·61-s + 12·67-s + 8·69-s + 16·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.258·15-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.840·51-s + 1.37·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.963·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.41536279286722, −12.99705253455175, −12.54404039714717, −12.27625881090128, −11.44262908498751, −11.06148776767471, −10.63715775205403, −10.39641804442980, −9.663939303627381, −9.265876680281551, −8.766044855610474, −8.038825861507354, −7.841526297450517, −6.967451504358164, −6.744032135454706, −6.057823110028078, −5.606565646606803, −5.197350454633243, −4.644596217283137, −3.910783781132209, −3.648536948073788, −2.551256818331945, −2.045431027452503, −1.853698249313133, −0.5564441374454756, 0,
0.5564441374454756, 1.853698249313133, 2.045431027452503, 2.551256818331945, 3.648536948073788, 3.910783781132209, 4.644596217283137, 5.197350454633243, 5.606565646606803, 6.057823110028078, 6.744032135454706, 6.967451504358164, 7.841526297450517, 8.038825861507354, 8.766044855610474, 9.265876680281551, 9.663939303627381, 10.39641804442980, 10.63715775205403, 11.06148776767471, 11.44262908498751, 12.27625881090128, 12.54404039714717, 12.99705253455175, 13.41536279286722
