| L(s) = 1 | − 5-s + 2·7-s + 4·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s − 2·35-s + 8·37-s + 8·43-s − 3·49-s + 6·53-s + 6·59-s − 2·61-s − 4·65-s + 4·67-s − 12·71-s + 10·73-s − 4·79-s − 12·83-s + 6·85-s − 12·89-s + 8·91-s + 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.31·37-s + 1.21·43-s − 3/7·49-s + 0.824·53-s + 0.781·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.450·79-s − 1.31·83-s + 0.650·85-s − 1.27·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.18082423315262, −13.56768709497715, −13.15672249456885, −12.79955857730104, −12.14295660734560, −11.47264384624217, −11.14401470844172, −10.94476749316538, −10.29964485923242, −9.589365073895468, −9.005655724075330, −8.587047863668166, −8.164808247349429, −7.681032131386023, −6.967611377294452, −6.551226239620947, −5.909827356237361, −5.447312546141591, −4.569501964043816, −4.220543043316389, −3.864920419650690, −2.902564913934536, −2.322566044100965, −1.651600502559343, −0.9088148584801605, 0,
0.9088148584801605, 1.651600502559343, 2.322566044100965, 2.902564913934536, 3.864920419650690, 4.220543043316389, 4.569501964043816, 5.447312546141591, 5.909827356237361, 6.551226239620947, 6.967611377294452, 7.681032131386023, 8.164808247349429, 8.587047863668166, 9.005655724075330, 9.589365073895468, 10.29964485923242, 10.94476749316538, 11.14401470844172, 11.47264384624217, 12.14295660734560, 12.79955857730104, 13.15672249456885, 13.56768709497715, 14.18082423315262
