| L(s) = 1 | − 5-s − 7-s − 11-s − 4·13-s + 6·17-s − 19-s + 9·23-s − 4·25-s + 11·31-s + 35-s + 37-s + 5·41-s + 2·43-s − 2·47-s + 49-s + 12·53-s + 55-s − 6·59-s + 4·65-s − 4·67-s + 13·71-s − 14·73-s + 77-s + 10·79-s + 8·83-s − 6·85-s − 3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.229·19-s + 1.87·23-s − 4/5·25-s + 1.97·31-s + 0.169·35-s + 0.164·37-s + 0.780·41-s + 0.304·43-s − 0.291·47-s + 1/7·49-s + 1.64·53-s + 0.134·55-s − 0.781·59-s + 0.496·65-s − 0.488·67-s + 1.54·71-s − 1.63·73-s + 0.113·77-s + 1.12·79-s + 0.878·83-s − 0.650·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.384346108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384346108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.608203759312194453797257789404, −8.645837641241072084676455533234, −7.73865828747377914951729213009, −7.24443827435431798091802148267, −6.22574815540833714031283437494, −5.27190466845715108765869362211, −4.48258397971251574389629801982, −3.33611157422544257194583862707, −2.53319485263396092629882859053, −0.835548470514578964121677318960,
0.835548470514578964121677318960, 2.53319485263396092629882859053, 3.33611157422544257194583862707, 4.48258397971251574389629801982, 5.27190466845715108765869362211, 6.22574815540833714031283437494, 7.24443827435431798091802148267, 7.73865828747377914951729213009, 8.645837641241072084676455533234, 9.608203759312194453797257789404
