| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s − 3·11-s + 4·13-s − 2·15-s + 8·17-s + 2·21-s + 2·23-s + 25-s + 4·27-s − 3·29-s + 7·31-s + 6·33-s − 35-s + 10·37-s − 8·39-s + 10·41-s − 2·43-s + 45-s − 4·47-s + 49-s − 16·51-s − 6·53-s − 3·55-s − 5·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.516·15-s + 1.94·17-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 0.557·29-s + 1.25·31-s + 1.04·33-s − 0.169·35-s + 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.304·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s − 2.24·51-s − 0.824·53-s − 0.404·55-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.70517431112962, −14.26238432400340, −13.64732508603613, −13.07104784424187, −12.79001372508815, −12.12943386959484, −11.73226204852860, −11.03066315565467, −10.76383382727821, −10.22544792045108, −9.629549183433085, −9.280706274641223, −8.290436954997372, −7.985863301081187, −7.353135899880981, −6.527823569083374, −6.137765025024953, −5.645476651000283, −5.327720916884735, −4.573868337381537, −3.902811182410382, −2.923768110404783, −2.768621235673685, −1.369603800805561, −1.002925887637282, 0,
1.002925887637282, 1.369603800805561, 2.768621235673685, 2.923768110404783, 3.902811182410382, 4.573868337381537, 5.327720916884735, 5.645476651000283, 6.137765025024953, 6.527823569083374, 7.353135899880981, 7.985863301081187, 8.290436954997372, 9.280706274641223, 9.629549183433085, 10.22544792045108, 10.76383382727821, 11.03066315565467, 11.73226204852860, 12.12943386959484, 12.79001372508815, 13.07104784424187, 13.64732508603613, 14.26238432400340, 14.70517431112962
