L(s) = 1 | − 4·5-s + 3·7-s − 4·11-s + 13-s + 4·17-s + 19-s + 11·25-s − 4·31-s − 12·35-s + 9·37-s + 8·43-s − 12·47-s + 2·49-s + 8·53-s + 16·55-s + 4·59-s + 5·61-s − 4·65-s − 11·67-s + 8·71-s + 73-s − 12·77-s + 5·79-s − 8·83-s − 16·85-s − 12·89-s + 3·91-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 1.13·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s + 11/5·25-s − 0.718·31-s − 2.02·35-s + 1.47·37-s + 1.21·43-s − 1.75·47-s + 2/7·49-s + 1.09·53-s + 2.15·55-s + 0.520·59-s + 0.640·61-s − 0.496·65-s − 1.34·67-s + 0.949·71-s + 0.117·73-s − 1.36·77-s + 0.562·79-s − 0.878·83-s − 1.73·85-s − 1.27·89-s + 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.96261257097759, −13.20215540405962, −12.89220673699620, −12.26669015787405, −11.91772241752183, −11.33290135598001, −11.07489239996102, −10.71314333372295, −9.998107180172627, −9.491677832164177, −8.594148833676968, −8.363115562219157, −7.846322951921087, −7.570597545737767, −7.191609812187707, −6.384524480573598, −5.550015076147820, −5.224160935296805, −4.613280774260459, −4.128558478616460, −3.606305204870360, −2.960051176391237, −2.412045229296271, −1.423728206081498, −0.8050290322611769, 0,
0.8050290322611769, 1.423728206081498, 2.412045229296271, 2.960051176391237, 3.606305204870360, 4.128558478616460, 4.613280774260459, 5.224160935296805, 5.550015076147820, 6.384524480573598, 7.191609812187707, 7.570597545737767, 7.846322951921087, 8.363115562219157, 8.594148833676968, 9.491677832164177, 9.998107180172627, 10.71314333372295, 11.07489239996102, 11.33290135598001, 11.91772241752183, 12.26669015787405, 12.89220673699620, 13.20215540405962, 13.96261257097759