L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 4·13-s + 15-s − 17-s − 21-s − 23-s + 25-s + 27-s − 7·29-s + 7·31-s − 35-s − 3·37-s − 4·39-s − 5·41-s − 12·43-s + 45-s + 6·47-s − 6·49-s − 51-s + 3·53-s − 7·59-s + 2·61-s − 63-s − 4·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 0.242·17-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.29·29-s + 1.25·31-s − 0.169·35-s − 0.493·37-s − 0.640·39-s − 0.780·41-s − 1.82·43-s + 0.149·45-s + 0.875·47-s − 6/7·49-s − 0.140·51-s + 0.412·53-s − 0.911·59-s + 0.256·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 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
−7.901431307888131872332445180809, −6.98557278512868231124352369313, −6.58783061535882375600027088347, −5.55688558181084733164749878001, −4.91208166952910542549184906368, −4.02848654637019068245012404822, −3.14507074768913777076622132099, −2.40664385260465043507061311613, −1.55491473208360045030397809683, 0,
1.55491473208360045030397809683, 2.40664385260465043507061311613, 3.14507074768913777076622132099, 4.02848654637019068245012404822, 4.91208166952910542549184906368, 5.55688558181084733164749878001, 6.58783061535882375600027088347, 6.98557278512868231124352369313, 7.901431307888131872332445180809