L(s) = 1 | − 3·5-s − 7-s + 3·11-s + 4·13-s − 2·19-s − 6·23-s + 4·25-s − 6·29-s + 5·31-s + 3·35-s − 2·37-s − 6·41-s + 10·43-s + 6·47-s − 6·49-s − 9·53-s − 9·55-s − 12·59-s − 8·61-s − 12·65-s − 14·67-s − 7·73-s − 3·77-s + 8·79-s + 3·83-s − 18·89-s − 4·91-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 0.377·7-s + 0.904·11-s + 1.10·13-s − 0.458·19-s − 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.898·31-s + 0.507·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 6/7·49-s − 1.23·53-s − 1.21·55-s − 1.56·59-s − 1.02·61-s − 1.48·65-s − 1.71·67-s − 0.819·73-s − 0.341·77-s + 0.900·79-s + 0.329·83-s − 1.90·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.853339056894194014756168928413, −8.104088285674841823256361524017, −7.46895104625204893367477999066, −6.47796475592373752838072674203, −5.89467970712550736408970851242, −4.43365380485938957073214377677, −3.92407328889816322387260499130, −3.12927788056823217499419163319, −1.52887349549650490954756981970, 0,
1.52887349549650490954756981970, 3.12927788056823217499419163319, 3.92407328889816322387260499130, 4.43365380485938957073214377677, 5.89467970712550736408970851242, 6.47796475592373752838072674203, 7.46895104625204893367477999066, 8.104088285674841823256361524017, 8.853339056894194014756168928413