| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 2·11-s + 2·15-s − 2·17-s + 4·21-s + 6·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 2·33-s + 8·35-s − 4·37-s + 2·41-s − 4·43-s − 2·45-s + 6·47-s + 9·49-s + 2·51-s + 2·53-s − 4·55-s − 12·59-s + 10·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.516·15-s − 0.485·17-s + 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s + 1.35·35-s − 0.657·37-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 1.56·59-s + 1.28·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.23045624483626796910538418393, −6.73504546613104419926843193212, −6.21402985368867299233181028685, −5.39570772751831333013763229821, −4.56409469882140149615399851948, −3.73872276927802553442634695645, −3.35817405055530012819687671302, −2.26882882735717096087619727984, −0.906053614823590096176464034696, 0,
0.906053614823590096176464034696, 2.26882882735717096087619727984, 3.35817405055530012819687671302, 3.73872276927802553442634695645, 4.56409469882140149615399851948, 5.39570772751831333013763229821, 6.21402985368867299233181028685, 6.73504546613104419926843193212, 7.23045624483626796910538418393
