| L(s) = 1 | − 0.254·2-s + 3-s − 1.93·4-s + 3.68·5-s − 0.254·6-s + 8-s + 9-s − 0.935·10-s + 11-s − 1.93·12-s − 2.18·13-s + 3.68·15-s + 3.61·16-s − 3.36·17-s − 0.254·18-s + 3.68·19-s − 7.12·20-s − 0.254·22-s + 24-s + 8.55·25-s + 0.556·26-s + 27-s + 10.0·29-s − 0.935·30-s + 8.37·31-s − 2.91·32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.179·2-s + 0.577·3-s − 0.967·4-s + 1.64·5-s − 0.103·6-s + 0.353·8-s + 0.333·9-s − 0.295·10-s + 0.301·11-s − 0.558·12-s − 0.607·13-s + 0.950·15-s + 0.904·16-s − 0.815·17-s − 0.0598·18-s + 0.844·19-s − 1.59·20-s − 0.0541·22-s + 0.204·24-s + 1.71·25-s + 0.109·26-s + 0.192·27-s + 1.86·29-s − 0.170·30-s + 1.50·31-s − 0.516·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.160169214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.160169214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.254T + 2T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 8.37T + 31T^{2} \) |
| 37 | \( 1 - 0.189T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 + 4.53T + 59T^{2} \) |
| 61 | \( 1 + 0.379T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 2.98T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 0.637T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−9.410871908325653588396944365211, −8.756582660688899738211093050144, −8.115309464806273868669255916328, −6.90911680735786991415459150149, −6.20454071047246781426443585209, −5.09765696056983483513151752259, −4.61660092151039590712923655750, −3.24696401115373763317777525918, −2.26498340245377359248150975724, −1.12353345687320657718825059207,
1.12353345687320657718825059207, 2.26498340245377359248150975724, 3.24696401115373763317777525918, 4.61660092151039590712923655750, 5.09765696056983483513151752259, 6.20454071047246781426443585209, 6.90911680735786991415459150149, 8.115309464806273868669255916328, 8.756582660688899738211093050144, 9.410871908325653588396944365211
