| L(s) = 1 | − 2·7-s − 11-s − 2·13-s + 2·19-s + 8·31-s − 2·37-s − 2·43-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s + 4·67-s − 2·73-s + 2·77-s − 10·79-s − 12·83-s + 6·89-s + 4·91-s − 14·97-s + 4·103-s − 12·107-s − 10·109-s + 6·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.458·19-s + 1.43·31-s − 0.328·37-s − 0.304·43-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.234·73-s + 0.227·77-s − 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s − 1.42·97-s + 0.394·103-s − 1.16·107-s − 0.957·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.15753456199974172411850260345, −6.78013055955743673329036825534, −5.94971852102651615678554628612, −5.30861911026750892030016967400, −4.57237734999796012817316169380, −3.75602998327925387915735454808, −2.95824340370193659910918518033, −2.34003650824815528773224494386, −1.12704153175159284767109286129, 0,
1.12704153175159284767109286129, 2.34003650824815528773224494386, 2.95824340370193659910918518033, 3.75602998327925387915735454808, 4.57237734999796012817316169380, 5.30861911026750892030016967400, 5.94971852102651615678554628612, 6.78013055955743673329036825534, 7.15753456199974172411850260345
