| L(s) = 1 | − 5-s − 7-s − 2·11-s − 5·17-s + 2·19-s − 2·23-s − 4·25-s − 10·29-s + 35-s + 5·37-s − 3·41-s − 7·43-s + 3·47-s + 49-s − 6·53-s + 2·55-s − 59-s − 6·61-s + 4·67-s − 8·71-s + 10·73-s + 2·77-s − 3·79-s + 13·83-s + 5·85-s + 6·89-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.21·17-s + 0.458·19-s − 0.417·23-s − 4/5·25-s − 1.85·29-s + 0.169·35-s + 0.821·37-s − 0.468·41-s − 1.06·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.269·55-s − 0.130·59-s − 0.768·61-s + 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.227·77-s − 0.337·79-s + 1.42·83-s + 0.542·85-s + 0.635·89-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.846079423573223773519846698993, −9.143301617654165472841605855665, −8.101884571922021378347981487463, −7.40939432753315854054987240240, −6.41203497630210983623322100128, −5.44331678883098011338394153195, −4.33915949405230466151992035191, −3.35741996591691126309031277005, −2.06159439572736423611160123888, 0,
2.06159439572736423611160123888, 3.35741996591691126309031277005, 4.33915949405230466151992035191, 5.44331678883098011338394153195, 6.41203497630210983623322100128, 7.40939432753315854054987240240, 8.101884571922021378347981487463, 9.143301617654165472841605855665, 9.846079423573223773519846698993
