L(s) = 1 | − 5-s + 7-s − 4·13-s + 17-s − 23-s + 25-s + 7·29-s − 7·31-s − 35-s − 3·37-s + 5·41-s + 12·43-s + 6·47-s − 6·49-s − 3·53-s − 7·59-s + 2·61-s + 4·65-s − 3·67-s + 3·71-s − 4·73-s − 12·79-s − 3·83-s − 85-s − 4·91-s − 18·97-s + 15·101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.10·13-s + 0.242·17-s − 0.208·23-s + 1/5·25-s + 1.29·29-s − 1.25·31-s − 0.169·35-s − 0.493·37-s + 0.780·41-s + 1.82·43-s + 0.875·47-s − 6/7·49-s − 0.412·53-s − 0.911·59-s + 0.256·61-s + 0.496·65-s − 0.366·67-s + 0.356·71-s − 0.468·73-s − 1.35·79-s − 0.329·83-s − 0.108·85-s − 0.419·91-s − 1.82·97-s + 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.53909444857105072183390307245, −6.93445380430343978086292410009, −6.04329833839234811953148275329, −5.32065957442136007816545767425, −4.60628508630792310870453674885, −4.01027823731459406954055453938, −3.01977423264983797980872895298, −2.30541335332653705343570253742, −1.21417796221568768521390254669, 0,
1.21417796221568768521390254669, 2.30541335332653705343570253742, 3.01977423264983797980872895298, 4.01027823731459406954055453938, 4.60628508630792310870453674885, 5.32065957442136007816545767425, 6.04329833839234811953148275329, 6.93445380430343978086292410009, 7.53909444857105072183390307245