| L(s) = 1 | + 3-s − 5-s − 4·7-s − 2·9-s − 6·11-s − 13-s − 15-s + 2·19-s − 4·21-s + 23-s + 25-s − 5·27-s + 9·29-s + 5·31-s − 6·33-s + 4·35-s + 2·37-s − 39-s − 9·41-s − 4·43-s + 2·45-s − 3·47-s + 9·49-s − 6·53-s + 6·55-s + 2·57-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s − 2/3·9-s − 1.80·11-s − 0.277·13-s − 0.258·15-s + 0.458·19-s − 0.872·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s + 0.898·31-s − 1.04·33-s + 0.676·35-s + 0.328·37-s − 0.160·39-s − 1.40·41-s − 0.609·43-s + 0.298·45-s − 0.437·47-s + 9/7·49-s − 0.824·53-s + 0.809·55-s + 0.264·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.33856671397141294446394116657, −9.849864980953179954431405276055, −8.679990191761162872669871605663, −8.000688813935122029953556753775, −6.99430430163915185465805017201, −5.92357863230048957860893165957, −4.79044752996760793533244334252, −3.17509326722106237588778190877, −2.78340880808103884678025920457, 0,
2.78340880808103884678025920457, 3.17509326722106237588778190877, 4.79044752996760793533244334252, 5.92357863230048957860893165957, 6.99430430163915185465805017201, 8.000688813935122029953556753775, 8.679990191761162872669871605663, 9.849864980953179954431405276055, 10.33856671397141294446394116657
