| L(s) = 1 | − 5-s − 2.82·7-s + 5.65·11-s − 2·13-s − 2·17-s − 2.82·23-s + 25-s − 6·29-s + 5.65·31-s + 2.82·35-s − 10·37-s − 2·41-s − 8.48·43-s + 2.82·47-s + 1.00·49-s − 6·53-s − 5.65·55-s − 11.3·59-s − 2·61-s + 2·65-s − 2.82·67-s − 5.65·71-s − 6·73-s − 16.0·77-s + 11.3·79-s − 2.82·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.06·7-s + 1.70·11-s − 0.554·13-s − 0.485·17-s − 0.589·23-s + 0.200·25-s − 1.11·29-s + 1.01·31-s + 0.478·35-s − 1.64·37-s − 0.312·41-s − 1.29·43-s + 0.412·47-s + 0.142·49-s − 0.824·53-s − 0.762·55-s − 1.47·59-s − 0.256·61-s + 0.248·65-s − 0.345·67-s − 0.671·71-s − 0.702·73-s − 1.82·77-s + 1.27·79-s − 0.310·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.206883926168802619577668862186, −8.459899785669851719256164874973, −7.35586860808036520391291120827, −6.66330364534288339791213305386, −6.07074588456278407762999436348, −4.77931011777860238276576962090, −3.86485250752345530176479745830, −3.14880370233397003691309117407, −1.69086024587047792801753188621, 0,
1.69086024587047792801753188621, 3.14880370233397003691309117407, 3.86485250752345530176479745830, 4.77931011777860238276576962090, 6.07074588456278407762999436348, 6.66330364534288339791213305386, 7.35586860808036520391291120827, 8.459899785669851719256164874973, 9.206883926168802619577668862186
