| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s − 2·9-s + 10-s + 4·11-s + 12-s − 15-s + 16-s + 2·18-s + 6·19-s − 20-s − 4·22-s + 7·23-s − 24-s + 25-s − 5·27-s − 9·29-s + 30-s − 4·31-s − 32-s + 4·33-s − 2·36-s − 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.471·18-s + 1.37·19-s − 0.223·20-s − 0.852·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-s − 1/3·36-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 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
−14.38576025589960, −13.78060190810445, −13.19877135193797, −12.77526963728922, −11.94252986967592, −11.70201763565544, −11.12756009448842, −10.96797165870584, −9.998564142870223, −9.504261352523406, −9.113948590330512, −8.838368018709623, −8.177206610616234, −7.579453993633478, −7.326680821712487, −6.620647875511615, −6.148420798663685, −5.298606950899215, −5.012310263314386, −3.907177392998378, −3.494014568474826, −3.106846326938541, −2.279423067157263, −1.568716677738616, −0.9266328028199933, 0,
0.9266328028199933, 1.568716677738616, 2.279423067157263, 3.106846326938541, 3.494014568474826, 3.907177392998378, 5.012310263314386, 5.298606950899215, 6.148420798663685, 6.620647875511615, 7.326680821712487, 7.579453993633478, 8.177206610616234, 8.838368018709623, 9.113948590330512, 9.504261352523406, 9.998564142870223, 10.96797165870584, 11.12756009448842, 11.70201763565544, 11.94252986967592, 12.77526963728922, 13.19877135193797, 13.78060190810445, 14.38576025589960
