| L(s) = 1 | − 5-s − 2·7-s − 2·13-s − 3·17-s + 5·19-s − 3·23-s + 25-s + 6·29-s − 5·31-s + 2·35-s − 2·37-s + 12·41-s + 8·43-s + 12·47-s − 3·49-s + 3·53-s + 6·59-s + 7·61-s + 2·65-s + 2·67-s − 12·71-s − 16·73-s + 79-s − 15·83-s + 3·85-s − 12·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.554·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s − 0.898·31-s + 0.338·35-s − 0.328·37-s + 1.87·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s + 0.412·53-s + 0.781·59-s + 0.896·61-s + 0.248·65-s + 0.244·67-s − 1.42·71-s − 1.87·73-s + 0.112·79-s − 1.64·83-s + 0.325·85-s − 1.27·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.20122350164184274533671544744, −7.04075563003671642478958301990, −5.93434957839749408997761213532, −5.52069146220157878802449830507, −4.41491108999081424469123801441, −3.99687375216374285606229667427, −2.96965737661263095689299043140, −2.43468093079104182755028491463, −1.09516607991712669897785898605, 0,
1.09516607991712669897785898605, 2.43468093079104182755028491463, 2.96965737661263095689299043140, 3.99687375216374285606229667427, 4.41491108999081424469123801441, 5.52069146220157878802449830507, 5.93434957839749408997761213532, 7.04075563003671642478958301990, 7.20122350164184274533671544744
