| L(s) = 1 | − 5-s − 2·7-s − 3·11-s − 4·13-s − 6·23-s + 25-s − 3·29-s − 5·31-s + 2·35-s + 8·37-s + 6·41-s + 4·43-s + 6·47-s − 3·49-s + 6·53-s + 3·55-s − 9·59-s − 7·61-s + 4·65-s − 2·67-s − 9·71-s − 4·73-s + 6·77-s + 7·79-s − 3·89-s + 8·91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.904·11-s − 1.10·13-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.898·31-s + 0.338·35-s + 1.31·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 0.404·55-s − 1.17·59-s − 0.896·61-s + 0.496·65-s − 0.244·67-s − 1.06·71-s − 0.468·73-s + 0.683·77-s + 0.787·79-s − 0.317·89-s + 0.838·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.99632427118721, −12.59710524436393, −12.11233091913063, −11.86064168190832, −11.09074228366384, −10.76379589221444, −10.28409207713773, −9.797594856177252, −9.306808910231029, −9.080513324117113, −8.204015598268427, −7.842588775462714, −7.429587743651539, −7.117670855826371, −6.331580316374678, −5.899786478155920, −5.494130787137616, −4.849385580930112, −4.251825798138298, −3.943056027539354, −3.157184330954316, −2.685864838060028, −2.271992925798624, −1.494212354356270, −0.5099285034285530, 0,
0.5099285034285530, 1.494212354356270, 2.271992925798624, 2.685864838060028, 3.157184330954316, 3.943056027539354, 4.251825798138298, 4.849385580930112, 5.494130787137616, 5.899786478155920, 6.331580316374678, 7.117670855826371, 7.429587743651539, 7.842588775462714, 8.204015598268427, 9.080513324117113, 9.306808910231029, 9.797594856177252, 10.28409207713773, 10.76379589221444, 11.09074228366384, 11.86064168190832, 12.11233091913063, 12.59710524436393, 12.99632427118721
