L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 3·11-s + 2·13-s + 15-s + 3·17-s − 2·19-s − 21-s + 25-s − 27-s − 3·29-s + 31-s + 3·33-s − 35-s + 37-s − 2·39-s + 9·41-s − 11·43-s − 45-s − 6·49-s − 3·51-s − 9·53-s + 3·55-s + 2·57-s + 6·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s − 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.557·29-s + 0.179·31-s + 0.522·33-s − 0.169·35-s + 0.164·37-s − 0.320·39-s + 1.40·41-s − 1.67·43-s − 0.149·45-s − 6/7·49-s − 0.420·51-s − 1.23·53-s + 0.404·55-s + 0.264·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.54639157547580503232112521288, −6.64822990041387487505048535721, −6.03639554084140272611685527545, −5.25690809165502152200666359824, −4.75917818805337181266027878549, −3.88822894374936851844416500793, −3.16122704824127312617534405145, −2.12446843606326016906390785364, −1.12027527288110619482046312281, 0,
1.12027527288110619482046312281, 2.12446843606326016906390785364, 3.16122704824127312617534405145, 3.88822894374936851844416500793, 4.75917818805337181266027878549, 5.25690809165502152200666359824, 6.03639554084140272611685527545, 6.64822990041387487505048535721, 7.54639157547580503232112521288