| L(s) = 1 | − 5·5-s + 4·7-s − 42·11-s + 20·13-s + 93·17-s − 59·19-s − 9·23-s + 25·25-s + 120·29-s − 47·31-s − 20·35-s − 262·37-s + 126·41-s + 178·43-s − 144·47-s − 327·49-s + 741·53-s + 210·55-s + 444·59-s + 221·61-s − 100·65-s + 538·67-s − 690·71-s − 1.12e3·73-s − 168·77-s − 665·79-s − 75·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.215·7-s − 1.15·11-s + 0.426·13-s + 1.32·17-s − 0.712·19-s − 0.0815·23-s + 1/5·25-s + 0.768·29-s − 0.272·31-s − 0.0965·35-s − 1.16·37-s + 0.479·41-s + 0.631·43-s − 0.446·47-s − 0.953·49-s + 1.92·53-s + 0.514·55-s + 0.979·59-s + 0.463·61-s − 0.190·65-s + 0.981·67-s − 1.15·71-s − 1.80·73-s − 0.248·77-s − 0.947·79-s − 0.0991·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 + 59 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 + 47 T + p^{3} T^{2} \) |
| 37 | \( 1 + 262 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 741 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 221 T + p^{3} T^{2} \) |
| 67 | \( 1 - 538 T + p^{3} T^{2} \) |
| 71 | \( 1 + 690 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1126 T + p^{3} T^{2} \) |
| 79 | \( 1 + 665 T + p^{3} T^{2} \) |
| 83 | \( 1 + 75 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1544 T + p^{3} T^{2} \) |
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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
−8.296080780529847386824352545721, −7.64999881352461416656693596212, −6.90473632520194198134227964847, −5.82702047148670919140756963211, −5.20210041328841900767858666969, −4.24054097178934030381802617249, −3.34245607507074733993194967598, −2.42134755849513429485289203930, −1.17308091506343467959325376583, 0,
1.17308091506343467959325376583, 2.42134755849513429485289203930, 3.34245607507074733993194967598, 4.24054097178934030381802617249, 5.20210041328841900767858666969, 5.82702047148670919140756963211, 6.90473632520194198134227964847, 7.64999881352461416656693596212, 8.296080780529847386824352545721
