| L(s) = 1 | − 2·3-s + 5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 2·19-s + 4·23-s + 25-s + 4·27-s + 10·29-s − 4·31-s + 8·33-s − 2·37-s − 4·39-s − 12·41-s + 4·43-s + 45-s − 4·47-s + 2·53-s − 4·55-s + 4·57-s − 10·59-s + 6·61-s + 2·65-s − 4·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s + 1.39·33-s − 0.328·37-s − 0.640·39-s − 1.87·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 0.274·53-s − 0.539·55-s + 0.529·57-s − 1.30·59-s + 0.768·61-s + 0.248·65-s − 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.230575364977733903972632960861, −7.13186266234597283620431631447, −6.51476717362374944138471434605, −5.84065513164095559271592430635, −5.13714452927980086262917176521, −4.67334494792962784426930508413, −3.35773186946108123811203955417, −2.46890536637459859099513059154, −1.21812233339541426962001417520, 0,
1.21812233339541426962001417520, 2.46890536637459859099513059154, 3.35773186946108123811203955417, 4.67334494792962784426930508413, 5.13714452927980086262917176521, 5.84065513164095559271592430635, 6.51476717362374944138471434605, 7.13186266234597283620431631447, 8.230575364977733903972632960861
