L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s + 13-s + 15-s + 17-s − 4·19-s + 25-s + 27-s − 2·29-s + 4·33-s + 6·37-s + 39-s − 6·41-s − 4·43-s + 45-s − 7·49-s + 51-s + 6·53-s + 4·55-s − 4·57-s + 4·59-s + 6·61-s + 65-s + 12·67-s − 16·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.768·61-s + 0.124·65-s + 1.46·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.639434547\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.639434547\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.94927360189301, −14.84966979056836, −14.30065016645842, −13.71364971371867, −13.15439237054233, −12.82947490326401, −12.00995995105913, −11.60305554032570, −10.95793568584061, −10.29590811755314, −9.767601183082672, −9.296654576073882, −8.651514000820659, −8.327840386938460, −7.535157292655575, −6.840003107596851, −6.406001451317247, −5.810461471549628, −4.997478136383548, −4.299262066705406, −3.711917246181186, −3.100338345468306, −2.166198355974687, −1.650309972171762, −0.7376957989916249,
0.7376957989916249, 1.650309972171762, 2.166198355974687, 3.100338345468306, 3.711917246181186, 4.299262066705406, 4.997478136383548, 5.810461471549628, 6.406001451317247, 6.840003107596851, 7.535157292655575, 8.327840386938460, 8.651514000820659, 9.296654576073882, 9.767601183082672, 10.29590811755314, 10.95793568584061, 11.60305554032570, 12.00995995105913, 12.82947490326401, 13.15439237054233, 13.71364971371867, 14.30065016645842, 14.84966979056836, 14.94927360189301