| L(s) = 1 | − 2-s − 0.601·3-s + 4-s − 1.03·5-s + 0.601·6-s − 8-s − 2.63·9-s + 1.03·10-s − 0.156·11-s − 0.601·12-s + 0.692·13-s + 0.623·15-s + 16-s + 7.16·17-s + 2.63·18-s + 3.66·19-s − 1.03·20-s + 0.156·22-s − 0.963·23-s + 0.601·24-s − 3.92·25-s − 0.692·26-s + 3.38·27-s − 2.22·29-s − 0.623·30-s + 2.92·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.347·3-s + 0.5·4-s − 0.463·5-s + 0.245·6-s − 0.353·8-s − 0.879·9-s + 0.328·10-s − 0.0472·11-s − 0.173·12-s + 0.191·13-s + 0.161·15-s + 0.250·16-s + 1.73·17-s + 0.621·18-s + 0.841·19-s − 0.231·20-s + 0.0334·22-s − 0.200·23-s + 0.122·24-s − 0.784·25-s − 0.135·26-s + 0.652·27-s − 0.413·29-s − 0.113·30-s + 0.525·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9566996673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9566996673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 101 | \( 1 + T \) |
| good | 3 | \( 1 + 0.601T + 3T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 11 | \( 1 + 0.156T + 11T^{2} \) |
| 13 | \( 1 - 0.692T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + 0.963T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 + 0.578T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 1.51T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.83965853956113580599051157390, −7.16576227526179200578557851377, −6.27614893780668059856833784967, −5.68134637275451722723470040045, −5.17763699255682775742061761812, −4.04671514108373369321920290377, −3.29167908685321286993231338590, −2.63946450914098792732191851653, −1.43263797794090613306424818936, −0.55831756050952367140181035185,
0.55831756050952367140181035185, 1.43263797794090613306424818936, 2.63946450914098792732191851653, 3.29167908685321286993231338590, 4.04671514108373369321920290377, 5.17763699255682775742061761812, 5.68134637275451722723470040045, 6.27614893780668059856833784967, 7.16576227526179200578557851377, 7.83965853956113580599051157390
