| L(s) = 1 | − 2-s + 3-s + 4-s + 1.72·5-s − 6-s − 0.765·7-s − 8-s + 9-s − 1.72·10-s − 5.74·11-s + 12-s + 2.78·13-s + 0.765·14-s + 1.72·15-s + 16-s + 0.587·17-s − 18-s − 19-s + 1.72·20-s − 0.765·21-s + 5.74·22-s + 0.319·23-s − 24-s − 2.02·25-s − 2.78·26-s + 27-s − 0.765·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.771·5-s − 0.408·6-s − 0.289·7-s − 0.353·8-s + 0.333·9-s − 0.545·10-s − 1.73·11-s + 0.288·12-s + 0.771·13-s + 0.204·14-s + 0.445·15-s + 0.250·16-s + 0.142·17-s − 0.235·18-s − 0.229·19-s + 0.385·20-s − 0.167·21-s + 1.22·22-s + 0.0666·23-s − 0.204·24-s − 0.404·25-s − 0.545·26-s + 0.192·27-s − 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 0.765T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 - 0.587T + 17T^{2} \) |
| 23 | \( 1 - 0.319T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 0.587T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 - 3.62T + 67T^{2} \) |
| 71 | \( 1 - 0.306T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 7.04T + 89T^{2} \) |
| 97 | \( 1 + 3.80T + 97T^{2} \) |
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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
−8.022639668867365952811485521513, −7.10593165664368819928969308000, −6.44201341678361985601774139929, −5.62183785888286201648103994341, −5.03911248209472116864263857170, −3.80281584853588764567234148388, −2.96065941314839741622563442666, −2.28303737631119782668856673813, −1.43731515676764821407867868451, 0,
1.43731515676764821407867868451, 2.28303737631119782668856673813, 2.96065941314839741622563442666, 3.80281584853588764567234148388, 5.03911248209472116864263857170, 5.62183785888286201648103994341, 6.44201341678361985601774139929, 7.10593165664368819928969308000, 8.022639668867365952811485521513
