| L(s) = 1 | − 2.23·2-s − 3.23·3-s + 3.00·4-s + 2·5-s + 7.23·6-s − 2.23·8-s + 7.47·9-s − 4.47·10-s − 11-s − 9.70·12-s + 1.23·13-s − 6.47·15-s − 0.999·16-s − 1.23·17-s − 16.7·18-s + 2.47·19-s + 6.00·20-s + 2.23·22-s − 6.47·23-s + 7.23·24-s − 25-s − 2.76·26-s − 14.4·27-s − 0.472·29-s + 14.4·30-s + 7.23·31-s + 6.70·32-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 1.86·3-s + 1.50·4-s + 0.894·5-s + 2.95·6-s − 0.790·8-s + 2.49·9-s − 1.41·10-s − 0.301·11-s − 2.80·12-s + 0.342·13-s − 1.67·15-s − 0.249·16-s − 0.299·17-s − 3.93·18-s + 0.567·19-s + 1.34·20-s + 0.476·22-s − 1.34·23-s + 1.47·24-s − 0.200·25-s − 0.542·26-s − 2.78·27-s − 0.0876·29-s + 2.64·30-s + 1.29·31-s + 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3836634795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3836634795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−10.62488177703540135650348344723, −9.987651779959547751517389905091, −9.472926087534246252774508044574, −8.147663586618887641096158432684, −7.18424524510303540791015816359, −6.26454448036419840097745758653, −5.69073997891943019562563874218, −4.44982373744121376579929552492, −2.02151533514478628667616299223, −0.77902122701161020743563112113,
0.77902122701161020743563112113, 2.02151533514478628667616299223, 4.44982373744121376579929552492, 5.69073997891943019562563874218, 6.26454448036419840097745758653, 7.18424524510303540791015816359, 8.147663586618887641096158432684, 9.472926087534246252774508044574, 9.987651779959547751517389905091, 10.62488177703540135650348344723
