| L(s) = 1 | − 2·11-s + 3·17-s − 19-s + 3·23-s − 4·29-s − 5·31-s − 10·37-s − 6·41-s + 6·43-s + 8·47-s − 7·49-s + 3·53-s + 5·61-s + 2·67-s − 2·71-s − 6·73-s − 11·79-s + 9·83-s − 10·89-s − 8·97-s + 12·101-s + 12·103-s − 4·107-s + 9·109-s − 6·113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s + 0.727·17-s − 0.229·19-s + 0.625·23-s − 0.742·29-s − 0.898·31-s − 1.64·37-s − 0.937·41-s + 0.914·43-s + 1.16·47-s − 49-s + 0.412·53-s + 0.640·61-s + 0.244·67-s − 0.237·71-s − 0.702·73-s − 1.23·79-s + 0.987·83-s − 1.05·89-s − 0.812·97-s + 1.19·101-s + 1.18·103-s − 0.386·107-s + 0.862·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66799247541723016901781805637, −7.26178345444521246986748164023, −6.39999755757971246516471452479, −5.48358742314242556721498817529, −5.10726254791347361757990363869, −4.00791804556625027557393879766, −3.30759479173743955199544638022, −2.37800474256130148896006176014, −1.37350549586058851111756287117, 0,
1.37350549586058851111756287117, 2.37800474256130148896006176014, 3.30759479173743955199544638022, 4.00791804556625027557393879766, 5.10726254791347361757990363869, 5.48358742314242556721498817529, 6.39999755757971246516471452479, 7.26178345444521246986748164023, 7.66799247541723016901781805637
