| L(s) = 1 | − 0.617·3-s − 2.13·5-s + 4.92·7-s − 2.61·9-s + 0.470·11-s − 1.22·13-s + 1.31·15-s + 5.19·17-s + 2.33·19-s − 3.04·21-s − 6.03·23-s − 0.449·25-s + 3.47·27-s − 8.75·31-s − 0.290·33-s − 10.5·35-s − 6.08·37-s + 0.757·39-s + 1.84·41-s + 4.01·43-s + 5.58·45-s − 8.17·47-s + 17.2·49-s − 3.20·51-s + 11.6·53-s − 1.00·55-s − 1.44·57-s + ⋯ |
| L(s) = 1 | − 0.356·3-s − 0.953·5-s + 1.86·7-s − 0.872·9-s + 0.141·11-s − 0.340·13-s + 0.340·15-s + 1.26·17-s + 0.536·19-s − 0.663·21-s − 1.25·23-s − 0.0899·25-s + 0.667·27-s − 1.57·31-s − 0.0505·33-s − 1.77·35-s − 1.00·37-s + 0.121·39-s + 0.288·41-s + 0.612·43-s + 0.832·45-s − 1.19·47-s + 2.46·49-s − 0.449·51-s + 1.60·53-s − 0.135·55-s − 0.191·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 0.617T + 3T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 0.470T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 6.03T + 23T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 4.01T + 43T^{2} \) |
| 47 | \( 1 + 8.17T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.274T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 0.912T + 67T^{2} \) |
| 71 | \( 1 + 5.15T + 71T^{2} \) |
| 73 | \( 1 + 2.18T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−7.56388865822541424463060700938, −7.34978092046460997960058740118, −5.97229790930901742303480097404, −5.45516459834377211145041277386, −4.85261311584496847781581464452, −4.03477214158736737036014264051, −3.33414784816187998092820330272, −2.16149388477465111432826323380, −1.25973150933715641796560706708, 0,
1.25973150933715641796560706708, 2.16149388477465111432826323380, 3.33414784816187998092820330272, 4.03477214158736737036014264051, 4.85261311584496847781581464452, 5.45516459834377211145041277386, 5.97229790930901742303480097404, 7.34978092046460997960058740118, 7.56388865822541424463060700938
