| L(s) = 1 | − 5-s + 2·13-s − 2·19-s + 25-s − 6·29-s + 8·31-s + 4·37-s − 6·41-s + 2·43-s − 6·47-s − 6·53-s − 12·59-s + 8·61-s − 2·65-s + 2·67-s − 6·71-s − 2·73-s + 16·79-s − 6·89-s + 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s − 0.824·53-s − 1.56·59-s + 1.02·61-s − 0.248·65-s + 0.244·67-s − 0.712·71-s − 0.234·73-s + 1.80·79-s − 0.635·89-s + 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.49439382623064, −13.27644849730775, −12.63055304623518, −12.24063619894471, −11.70670598182109, −11.10006330528940, −11.03397861944859, −10.21716200943764, −9.840785766305583, −9.308673003653411, −8.612577414469318, −8.429188737732475, −7.687723183054449, −7.452145362743334, −6.622977015250715, −6.275856973819153, −5.816103414401025, −4.947123552369656, −4.695005922722396, −3.963466165762091, −3.489166486344579, −2.938332309220514, −2.197482602302538, −1.554233081712647, −0.8054575590821579, 0,
0.8054575590821579, 1.554233081712647, 2.197482602302538, 2.938332309220514, 3.489166486344579, 3.963466165762091, 4.695005922722396, 4.947123552369656, 5.816103414401025, 6.275856973819153, 6.622977015250715, 7.452145362743334, 7.687723183054449, 8.429188737732475, 8.612577414469318, 9.308673003653411, 9.840785766305583, 10.21716200943764, 11.03397861944859, 11.10006330528940, 11.70670598182109, 12.24063619894471, 12.63055304623518, 13.27644849730775, 13.49439382623064
