| L(s) = 1 | + 5-s − 2·7-s + 11-s + 2·13-s − 8·17-s − 2·19-s − 8·23-s + 25-s − 2·35-s + 2·37-s + 6·43-s − 8·47-s − 3·49-s − 6·53-s + 55-s + 4·59-s + 10·61-s + 2·65-s − 12·67-s − 8·71-s + 10·73-s − 2·77-s − 14·79-s − 4·83-s − 8·85-s − 10·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s − 1.94·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.338·35-s + 0.328·37-s + 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.227·77-s − 1.57·79-s − 0.439·83-s − 0.867·85-s − 1.05·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.815402478670686463760440783979, −8.169017980160617904405062114394, −6.99344125797315299349020742533, −6.36132389521375291447967401252, −5.83564097890541512904268284014, −4.55143367824604237806042969574, −3.87155450207857290945723404660, −2.69260207982730188842610789668, −1.74063382717034329602598274446, 0,
1.74063382717034329602598274446, 2.69260207982730188842610789668, 3.87155450207857290945723404660, 4.55143367824604237806042969574, 5.83564097890541512904268284014, 6.36132389521375291447967401252, 6.99344125797315299349020742533, 8.169017980160617904405062114394, 8.815402478670686463760440783979
