| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 6·13-s − 2·15-s + 17-s + 21-s + 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 2·35-s + 10·37-s − 6·39-s − 6·41-s − 12·43-s − 2·45-s + 49-s + 51-s − 10·53-s + 8·59-s + 6·61-s + 63-s + 12·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.242·17-s + 0.218·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.338·35-s + 1.64·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 1/7·49-s + 0.140·51-s − 1.37·53-s + 1.04·59-s + 0.768·61-s + 0.125·63-s + 1.48·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−7.87926904976884309319145553824, −7.19515866090246264939530945112, −6.63441653407793169712897922103, −5.37292810110458713350651257174, −4.77235890347525183836188577110, −4.11402402903198192748250030289, −3.14603790489072703936478308725, −2.53767352748806757689866707676, −1.36698861038099470242315828550, 0,
1.36698861038099470242315828550, 2.53767352748806757689866707676, 3.14603790489072703936478308725, 4.11402402903198192748250030289, 4.77235890347525183836188577110, 5.37292810110458713350651257174, 6.63441653407793169712897922103, 7.19515866090246264939530945112, 7.87926904976884309319145553824
