| L(s) = 1 | + 2-s − 3-s + 4-s + 0.732·5-s − 6-s − 7-s + 8-s + 9-s + 0.732·10-s − 3.73·11-s − 12-s − 14-s − 0.732·15-s + 16-s + 0.267·17-s + 18-s + 4.46·19-s + 0.732·20-s + 21-s − 3.73·22-s + 3.46·23-s − 24-s − 4.46·25-s − 27-s − 28-s − 3·29-s − 0.732·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.327·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.231·10-s − 1.12·11-s − 0.288·12-s − 0.267·14-s − 0.189·15-s + 0.250·16-s + 0.0649·17-s + 0.235·18-s + 1.02·19-s + 0.163·20-s + 0.218·21-s − 0.795·22-s + 0.722·23-s − 0.204·24-s − 0.892·25-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 17 | \( 1 - 0.267T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 0.732T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 0.732T + 97T^{2} \) |
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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
−7.58145509631131023494676008189, −6.73199870497150311580813820600, −6.01550537557317169249446694808, −5.39864750948483400149937065416, −4.99880740223038422986682391697, −3.98807138854505203231244587961, −3.21240106183492532178608480583, −2.41179960947626628687517788851, −1.38733439384469632512183222708, 0,
1.38733439384469632512183222708, 2.41179960947626628687517788851, 3.21240106183492532178608480583, 3.98807138854505203231244587961, 4.99880740223038422986682391697, 5.39864750948483400149937065416, 6.01550537557317169249446694808, 6.73199870497150311580813820600, 7.58145509631131023494676008189
