| L(s) = 1 | + 3-s − 5-s − 2·7-s − 2·9-s − 11-s − 4·13-s − 15-s − 2·17-s − 2·21-s − 23-s − 4·25-s − 5·27-s + 7·31-s − 33-s + 2·35-s − 3·37-s − 4·39-s − 8·41-s + 6·43-s + 2·45-s + 8·47-s − 3·49-s − 2·51-s + 6·53-s + 55-s − 5·59-s − 12·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.436·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.25·31-s − 0.174·33-s + 0.338·35-s − 0.493·37-s − 0.640·39-s − 1.24·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 0.134·55-s − 0.650·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.884479599741513069462979932336, −9.166677969226674814621743788738, −8.260925990474913957804239520396, −7.52911496869612636142589120275, −6.54856398727522524336813919579, −5.49806099701646720543351753580, −4.32235737580261385936793495239, −3.21061399039481794187688129095, −2.34256559818964998262222027168, 0,
2.34256559818964998262222027168, 3.21061399039481794187688129095, 4.32235737580261385936793495239, 5.49806099701646720543351753580, 6.54856398727522524336813919579, 7.52911496869612636142589120275, 8.260925990474913957804239520396, 9.166677969226674814621743788738, 9.884479599741513069462979932336
