| L(s) = 1 | − 0.745·2-s − 0.549·3-s − 1.44·4-s + 3.44·5-s + 0.410·6-s + 5.17·7-s + 2.56·8-s − 2.69·9-s − 2.56·10-s − 1.31·11-s + 0.793·12-s − 3.54·13-s − 3.86·14-s − 1.89·15-s + 0.970·16-s − 2.15·17-s + 2.01·18-s + 5.13·19-s − 4.97·20-s − 2.84·21-s + 0.980·22-s − 3.52·23-s − 1.41·24-s + 6.85·25-s + 2.64·26-s + 3.13·27-s − 7.47·28-s + ⋯ |
| L(s) = 1 | − 0.527·2-s − 0.317·3-s − 0.721·4-s + 1.54·5-s + 0.167·6-s + 1.95·7-s + 0.908·8-s − 0.899·9-s − 0.812·10-s − 0.396·11-s + 0.229·12-s − 0.982·13-s − 1.03·14-s − 0.488·15-s + 0.242·16-s − 0.522·17-s + 0.474·18-s + 1.17·19-s − 1.11·20-s − 0.620·21-s + 0.209·22-s − 0.735·23-s − 0.288·24-s + 1.37·25-s + 0.518·26-s + 0.602·27-s − 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7970984885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7970984885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 89 | \( 1 - T \) |
| good | 2 | \( 1 + 0.745T + 2T^{2} \) |
| 3 | \( 1 + 0.549T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 5.17T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 + 9.30T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 - 2.80T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 4.44T + 61T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 - 0.666T + 83T^{2} \) |
| 97 | \( 1 - 5.36T + 97T^{2} \) |
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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
−14.10543548581542141727301920343, −13.41161188297091128382565921622, −11.80945838265680740951693655682, −10.73265760358139432249830184986, −9.716274834393394874869932818892, −8.689788781707687792554591580430, −7.61779099427155181338745850111, −5.53222976586399877342488961564, −4.95283656456386803345642958433, −1.91341662104729503875907175725,
1.91341662104729503875907175725, 4.95283656456386803345642958433, 5.53222976586399877342488961564, 7.61779099427155181338745850111, 8.689788781707687792554591580430, 9.716274834393394874869932818892, 10.73265760358139432249830184986, 11.80945838265680740951693655682, 13.41161188297091128382565921622, 14.10543548581542141727301920343
