| L(s) = 1 | + 2.28·2-s + 3.22·4-s − 5-s + 2.51·7-s + 2.80·8-s − 2.28·10-s + 11-s + 6.05·13-s + 5.76·14-s − 0.0392·16-s − 4.97·17-s − 7.02·19-s − 3.22·20-s + 2.28·22-s + 4.45·23-s + 25-s + 13.8·26-s + 8.13·28-s + 0.921·29-s + 3.03·31-s − 5.70·32-s − 11.3·34-s − 2.51·35-s − 3.49·37-s − 16.0·38-s − 2.80·40-s − 10.0·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s − 0.447·5-s + 0.952·7-s + 0.992·8-s − 0.722·10-s + 0.301·11-s + 1.67·13-s + 1.53·14-s − 0.00981·16-s − 1.20·17-s − 1.61·19-s − 0.721·20-s + 0.487·22-s + 0.928·23-s + 0.200·25-s + 2.71·26-s + 1.53·28-s + 0.171·29-s + 0.545·31-s − 1.00·32-s − 1.95·34-s − 0.425·35-s − 0.574·37-s − 2.60·38-s − 0.443·40-s − 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.417491590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417491590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 0.921T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 8.69T + 67T^{2} \) |
| 71 | \( 1 - 1.54T + 71T^{2} \) |
| 73 | \( 1 - 6.05T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 8.50T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−11.16974359454072944976660186240, −10.71421678295840862449871639226, −8.844732705315567740520444723530, −8.277403358776855777713178053940, −6.79591229519875681372710899646, −6.25239210996437674214797573033, −4.96322108299691404875600581196, −4.30061259606992361945299167373, −3.36804369237588573730318615821, −1.86928691673103648163089535823,
1.86928691673103648163089535823, 3.36804369237588573730318615821, 4.30061259606992361945299167373, 4.96322108299691404875600581196, 6.25239210996437674214797573033, 6.79591229519875681372710899646, 8.277403358776855777713178053940, 8.844732705315567740520444723530, 10.71421678295840862449871639226, 11.16974359454072944976660186240
