| L(s) = 1 | − 2-s + 1.62·3-s + 4-s − 1.42·5-s − 1.62·6-s − 2.34·7-s − 8-s − 0.348·9-s + 1.42·10-s − 2.52·11-s + 1.62·12-s − 1.30·13-s + 2.34·14-s − 2.31·15-s + 16-s − 0.938·17-s + 0.348·18-s − 4.18·19-s − 1.42·20-s − 3.82·21-s + 2.52·22-s + 9.28·23-s − 1.62·24-s − 2.98·25-s + 1.30·26-s − 5.45·27-s − 2.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.940·3-s + 0.5·4-s − 0.635·5-s − 0.664·6-s − 0.887·7-s − 0.353·8-s − 0.116·9-s + 0.449·10-s − 0.760·11-s + 0.470·12-s − 0.361·13-s + 0.627·14-s − 0.597·15-s + 0.250·16-s − 0.227·17-s + 0.0821·18-s − 0.960·19-s − 0.317·20-s − 0.833·21-s + 0.537·22-s + 1.93·23-s − 0.332·24-s − 0.596·25-s + 0.255·26-s − 1.04·27-s − 0.443·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7991930024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7991930024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 0.938T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 9.28T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 37 | \( 1 - 4.90T + 37T^{2} \) |
| 41 | \( 1 + 0.744T + 41T^{2} \) |
| 43 | \( 1 - 2.99T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 + 2.88T + 59T^{2} \) |
| 61 | \( 1 + 2.81T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 0.217T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
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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
−8.135678806303489675996326584244, −7.49778396713688998350891814417, −7.02021742083401980698468449374, −6.08808250329111773316615273078, −5.30144955812376273457753449453, −4.17238064688032758716446541516, −3.40448647214319468819662114465, −2.73693055368900089781782962665, −2.05171646786136574638582824636, −0.46454854633275116574171908781,
0.46454854633275116574171908781, 2.05171646786136574638582824636, 2.73693055368900089781782962665, 3.40448647214319468819662114465, 4.17238064688032758716446541516, 5.30144955812376273457753449453, 6.08808250329111773316615273078, 7.02021742083401980698468449374, 7.49778396713688998350891814417, 8.135678806303489675996326584244
