| L(s) = 1 | − 2·7-s − 6·11-s + 13-s + 19-s − 6·23-s − 3·29-s − 8·31-s − 37-s − 9·41-s − 8·43-s + 3·47-s − 3·49-s + 3·53-s + 6·59-s − 10·61-s + 13·67-s − 9·71-s − 4·73-s + 12·77-s − 11·79-s − 12·83-s − 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.80·11-s + 0.277·13-s + 0.229·19-s − 1.25·23-s − 0.557·29-s − 1.43·31-s − 0.164·37-s − 1.40·41-s − 1.21·43-s + 0.437·47-s − 3/7·49-s + 0.412·53-s + 0.781·59-s − 1.28·61-s + 1.58·67-s − 1.06·71-s − 0.468·73-s + 1.36·77-s − 1.23·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.14882859524860, −14.68287466476768, −13.96633503735363, −13.36806435457849, −13.22618734363206, −12.56797298803188, −12.17881447600634, −11.40536522787236, −10.98413612489032, −10.31511556340355, −9.960382427997048, −9.567503846165839, −8.621039186899686, −8.388422120543077, −7.641672860465429, −7.215287679646803, −6.608117316965534, −5.802487236170437, −5.515609239590675, −4.900870312024886, −4.067489478408799, −3.445543888047847, −2.909272068602295, −2.179464893065863, −1.490370937314112, 0, 0,
1.490370937314112, 2.179464893065863, 2.909272068602295, 3.445543888047847, 4.067489478408799, 4.900870312024886, 5.515609239590675, 5.802487236170437, 6.608117316965534, 7.215287679646803, 7.641672860465429, 8.388422120543077, 8.621039186899686, 9.567503846165839, 9.960382427997048, 10.31511556340355, 10.98413612489032, 11.40536522787236, 12.17881447600634, 12.56797298803188, 13.22618734363206, 13.36806435457849, 13.96633503735363, 14.68287466476768, 15.14882859524860
