| L(s) = 1 | + 2·5-s + 2·13-s + 2·17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 8·31-s + 6·37-s − 6·41-s − 4·43-s − 7·49-s + 2·53-s − 4·59-s + 2·61-s + 4·65-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s − 4·83-s + 4·85-s + 6·89-s + 8·95-s + 2·97-s − 18·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.433·85-s + 0.635·89-s + 0.820·95-s + 0.203·97-s − 1.79·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.003209511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003209511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.83654723068629114933839597949, −6.92796334648786182474479663711, −6.40725479078387541676599821855, −5.70681982833013824339552087978, −5.05736853539339825666279063330, −4.38461660538881471089251752313, −3.21029387756548432515894957212, −2.80691305312846013255691010566, −1.61011393533291281462985460586, −0.918192672118257559483050015003,
0.918192672118257559483050015003, 1.61011393533291281462985460586, 2.80691305312846013255691010566, 3.21029387756548432515894957212, 4.38461660538881471089251752313, 5.05736853539339825666279063330, 5.70681982833013824339552087978, 6.40725479078387541676599821855, 6.92796334648786182474479663711, 7.83654723068629114933839597949
