| L(s) = 1 | + 0.291·2-s + 3.14·3-s − 1.91·4-s + 0.915·6-s − 1.20·7-s − 1.14·8-s + 6.86·9-s + 3.65·11-s − 6.01·12-s + 0.859·13-s − 0.351·14-s + 3.49·16-s + 6.72·17-s + 2·18-s − 1.51·19-s − 3.78·21-s + 1.06·22-s − 23-s − 3.58·24-s + 0.250·26-s + 12.1·27-s + 2.31·28-s + 0.548·29-s − 5.99·31-s + 3.30·32-s + 11.4·33-s + 1.95·34-s + ⋯ |
| L(s) = 1 | + 0.206·2-s + 1.81·3-s − 0.957·4-s + 0.373·6-s − 0.456·7-s − 0.403·8-s + 2.28·9-s + 1.10·11-s − 1.73·12-s + 0.238·13-s − 0.0939·14-s + 0.874·16-s + 1.63·17-s + 0.471·18-s − 0.348·19-s − 0.826·21-s + 0.227·22-s − 0.208·23-s − 0.731·24-s + 0.0491·26-s + 2.33·27-s + 0.436·28-s + 0.101·29-s − 1.07·31-s + 0.583·32-s + 1.99·33-s + 0.335·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460091109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460091109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.291T + 2T^{2} \) |
| 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 - 0.859T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 29 | \( 1 - 0.548T + 29T^{2} \) |
| 31 | \( 1 + 5.99T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.17T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 - 0.0700T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 - 4.96T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−10.17227559076433503223035782281, −9.668129478573489215165131540627, −8.933824721317513446752561914100, −8.306249017619696127757590630996, −7.44783128667451917948868963626, −6.25060654131136835163175550793, −4.81127330466023954237722377944, −3.61703720831216474199518740096, −3.32703472465273216772733312966, −1.56084702746934491231264930856,
1.56084702746934491231264930856, 3.32703472465273216772733312966, 3.61703720831216474199518740096, 4.81127330466023954237722377944, 6.25060654131136835163175550793, 7.44783128667451917948868963626, 8.306249017619696127757590630996, 8.933824721317513446752561914100, 9.668129478573489215165131540627, 10.17227559076433503223035782281
