| L(s) = 1 | − 5-s − 4·7-s + 2·11-s + 2·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s + 4·29-s − 31-s + 4·35-s − 2·37-s − 4·43-s + 12·47-s + 9·49-s + 6·53-s − 2·55-s + 8·59-s − 14·61-s − 2·65-s − 2·67-s − 10·71-s − 8·77-s − 4·79-s + 2·85-s + 6·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s − 0.179·31-s + 0.676·35-s − 0.328·37-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s + 1.04·59-s − 1.79·61-s − 0.248·65-s − 0.244·67-s − 1.18·71-s − 0.911·77-s − 0.450·79-s + 0.216·85-s + 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.96621477653938, −13.54017218057608, −13.12815971142902, −12.68269782496640, −12.19427867707558, −11.76653625473745, −11.07941567014075, −10.61942973759075, −10.28616666389442, −9.526845069408157, −9.033418649542571, −8.785762912229314, −8.224286531640344, −7.330007404725857, −6.948338374827599, −6.553635126834483, −6.051533474209351, −5.452486720473217, −4.653743953201009, −4.090711282417015, −3.651828956804413, −2.951848674566388, −2.587988107007430, −1.532558822458179, −0.7853655760547264, 0,
0.7853655760547264, 1.532558822458179, 2.587988107007430, 2.951848674566388, 3.651828956804413, 4.090711282417015, 4.653743953201009, 5.452486720473217, 6.051533474209351, 6.553635126834483, 6.948338374827599, 7.330007404725857, 8.224286531640344, 8.785762912229314, 9.033418649542571, 9.526845069408157, 10.28616666389442, 10.61942973759075, 11.07941567014075, 11.76653625473745, 12.19427867707558, 12.68269782496640, 13.12815971142902, 13.54017218057608, 13.96621477653938
