| L(s) = 1 | + 3-s − 0.732·5-s − 7-s + 9-s + 1.46·13-s − 0.732·15-s − 4.73·17-s + 19-s − 21-s − 4.46·25-s + 27-s + 7.66·29-s − 2·31-s + 0.732·35-s − 10·37-s + 1.46·39-s + 4.92·41-s + 8.92·43-s − 0.732·45-s − 5.66·47-s + 49-s − 4.73·51-s − 3.26·53-s + 57-s − 8·59-s − 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.327·5-s − 0.377·7-s + 0.333·9-s + 0.406·13-s − 0.189·15-s − 1.14·17-s + 0.229·19-s − 0.218·21-s − 0.892·25-s + 0.192·27-s + 1.42·29-s − 0.359·31-s + 0.123·35-s − 1.64·37-s + 0.234·39-s + 0.769·41-s + 1.36·43-s − 0.109·45-s − 0.825·47-s + 0.142·49-s − 0.662·51-s − 0.448·53-s + 0.132·57-s − 1.04·59-s − 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.75097658324454281523041524250, −6.98275522849505534705897299672, −6.40679755153958001358303879040, −5.58246225065300263197115831338, −4.58988385864815230136055099571, −3.99576836336005103601443813960, −3.18717065166689029113600976216, −2.41686005401268040895230772411, −1.39529355434643186972089229099, 0,
1.39529355434643186972089229099, 2.41686005401268040895230772411, 3.18717065166689029113600976216, 3.99576836336005103601443813960, 4.58988385864815230136055099571, 5.58246225065300263197115831338, 6.40679755153958001358303879040, 6.98275522849505534705897299672, 7.75097658324454281523041524250
