| L(s) = 1 | + 3·5-s − 11-s + 13-s + 4·17-s + 3·19-s + 6·23-s + 4·25-s − 5·29-s + 4·31-s + 37-s + 6·41-s + 4·43-s + 7·47-s − 2·53-s − 3·55-s − 59-s + 2·61-s + 3·65-s + 7·67-s − 10·71-s − 73-s − 16·79-s + 8·83-s + 12·85-s + 14·89-s + 9·95-s − 16·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.301·11-s + 0.277·13-s + 0.970·17-s + 0.688·19-s + 1.25·23-s + 4/5·25-s − 0.928·29-s + 0.718·31-s + 0.164·37-s + 0.937·41-s + 0.609·43-s + 1.02·47-s − 0.274·53-s − 0.404·55-s − 0.130·59-s + 0.256·61-s + 0.372·65-s + 0.855·67-s − 1.18·71-s − 0.117·73-s − 1.80·79-s + 0.878·83-s + 1.30·85-s + 1.48·89-s + 0.923·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19404 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.569388338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569388338\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.81840864714064, −15.00361178041767, −14.59624988703241, −13.95687215750688, −13.58561890734629, −13.02633285262562, −12.55618612808328, −11.89169776723789, −11.13415203248018, −10.72210346634824, −9.971750743080703, −9.658031531699330, −9.061817035385561, −8.507070533075336, −7.573604733227497, −7.270073622092576, −6.338209499700840, −5.818140167631127, −5.389660006475541, −4.732450803007766, −3.805964625454412, −2.982924012676732, −2.437547152431815, −1.502335861949035, −0.8442808136708142,
0.8442808136708142, 1.502335861949035, 2.437547152431815, 2.982924012676732, 3.805964625454412, 4.732450803007766, 5.389660006475541, 5.818140167631127, 6.338209499700840, 7.270073622092576, 7.573604733227497, 8.507070533075336, 9.061817035385561, 9.658031531699330, 9.971750743080703, 10.72210346634824, 11.13415203248018, 11.89169776723789, 12.55618612808328, 13.02633285262562, 13.58561890734629, 13.95687215750688, 14.59624988703241, 15.00361178041767, 15.81840864714064
