| L(s) = 1 | − 5-s + 3·7-s − 2·13-s − 6·17-s + 5·19-s + 4·23-s + 25-s − 6·29-s + 5·31-s − 3·35-s − 11·37-s − 4·41-s + 4·43-s − 6·47-s + 2·49-s + 8·53-s − 2·59-s + 61-s + 2·65-s + 7·67-s − 10·71-s − 11·73-s − 79-s + 10·83-s + 6·85-s + 14·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.554·13-s − 1.45·17-s + 1.14·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.898·31-s − 0.507·35-s − 1.80·37-s − 0.624·41-s + 0.609·43-s − 0.875·47-s + 2/7·49-s + 1.09·53-s − 0.260·59-s + 0.128·61-s + 0.248·65-s + 0.855·67-s − 1.18·71-s − 1.28·73-s − 0.112·79-s + 1.09·83-s + 0.650·85-s + 1.48·89-s − 0.628·91-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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.12636471438472, −13.66688221305267, −13.24841991249083, −12.65615329705550, −12.02147857844819, −11.58343674391326, −11.36638717024578, −10.69523626818537, −10.33793846438601, −9.575753969695416, −9.076264889926215, −8.584716039543947, −8.157873877694013, −7.488102021694620, −7.116923889991874, −6.694734753719993, −5.792279793655095, −5.263481197681117, −4.725305106614791, −4.430255922404482, −3.564291933861182, −3.063603074851116, −2.203500159281768, −1.727702436633653, −0.8956699450148909, 0,
0.8956699450148909, 1.727702436633653, 2.203500159281768, 3.063603074851116, 3.564291933861182, 4.430255922404482, 4.725305106614791, 5.263481197681117, 5.792279793655095, 6.694734753719993, 7.116923889991874, 7.488102021694620, 8.157873877694013, 8.584716039543947, 9.076264889926215, 9.575753969695416, 10.33793846438601, 10.69523626818537, 11.36638717024578, 11.58343674391326, 12.02147857844819, 12.65615329705550, 13.24841991249083, 13.66688221305267, 14.12636471438472
