| L(s) = 1 | − 0.262·2-s − 1.93·4-s − 0.704·7-s + 1.03·8-s + 11-s − 3.82·13-s + 0.185·14-s + 3.59·16-s − 7.15·17-s + 7.33·19-s − 0.262·22-s + 5.86·23-s + 1.00·26-s + 1.36·28-s − 8.97·29-s − 0.590·31-s − 3.00·32-s + 1.88·34-s − 10.4·37-s − 1.92·38-s + 7.92·41-s − 3.29·43-s − 1.93·44-s − 1.53·46-s + 3.64·47-s − 6.50·49-s + 7.37·52-s + ⋯ |
| L(s) = 1 | − 0.185·2-s − 0.965·4-s − 0.266·7-s + 0.365·8-s + 0.301·11-s − 1.05·13-s + 0.0494·14-s + 0.897·16-s − 1.73·17-s + 1.68·19-s − 0.0560·22-s + 1.22·23-s + 0.196·26-s + 0.257·28-s − 1.66·29-s − 0.106·31-s − 0.531·32-s + 0.322·34-s − 1.71·37-s − 0.312·38-s + 1.23·41-s − 0.502·43-s − 0.291·44-s − 0.227·46-s + 0.531·47-s − 0.929·49-s + 1.02·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9639257287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9639257287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.262T + 2T^{2} \) |
| 7 | \( 1 + 0.704T + 7T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 - 7.33T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.590T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 3.82T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 - 7.05T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−9.116909345514854589185362958009, −8.332330607883632116659329163310, −7.25733764998635422961021872800, −6.94109924398733212625661120082, −5.52984827909774754932450117295, −5.07351536995454656999099382870, −4.13217400446823671570510380719, −3.30491123243695141938213082595, −2.08677333745485357034507104781, −0.63818086858181527632687224966,
0.63818086858181527632687224966, 2.08677333745485357034507104781, 3.30491123243695141938213082595, 4.13217400446823671570510380719, 5.07351536995454656999099382870, 5.52984827909774754932450117295, 6.94109924398733212625661120082, 7.25733764998635422961021872800, 8.332330607883632116659329163310, 9.116909345514854589185362958009
