| L(s) = 1 | + 0.968·2-s − 1.06·4-s − 2.59·5-s + 3.83·7-s − 2.96·8-s − 2.51·10-s − 0.279·11-s − 4.21·13-s + 3.71·14-s − 0.748·16-s − 7.81·17-s + 5.87·19-s + 2.75·20-s − 0.270·22-s − 3.65·23-s + 1.74·25-s − 4.08·26-s − 4.07·28-s − 0.344·29-s + 6.17·31-s + 5.20·32-s − 7.56·34-s − 9.95·35-s − 3.88·37-s + 5.69·38-s + 7.70·40-s − 2.01·41-s + ⋯ |
| L(s) = 1 | + 0.684·2-s − 0.530·4-s − 1.16·5-s + 1.44·7-s − 1.04·8-s − 0.795·10-s − 0.0842·11-s − 1.16·13-s + 0.992·14-s − 0.187·16-s − 1.89·17-s + 1.34·19-s + 0.616·20-s − 0.0577·22-s − 0.761·23-s + 0.348·25-s − 0.801·26-s − 0.769·28-s − 0.0639·29-s + 1.10·31-s + 0.920·32-s − 1.29·34-s − 1.68·35-s − 0.639·37-s + 0.923·38-s + 1.21·40-s − 0.314·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370055376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370055376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 0.968T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 0.279T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.344T + 29T^{2} \) |
| 31 | \( 1 - 6.17T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 - 3.28T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.174594550583308790631073415644, −7.40982171108430778121626043824, −6.76913046733328886161902253858, −5.60557491030811541657963881620, −4.97148469024484853447742440429, −4.45106195802573282954256297063, −3.98740275493976011333141330458, −2.94614297244641165275296074161, −2.01337324876070144619223800593, −0.54189786840997052419528426548,
0.54189786840997052419528426548, 2.01337324876070144619223800593, 2.94614297244641165275296074161, 3.98740275493976011333141330458, 4.45106195802573282954256297063, 4.97148469024484853447742440429, 5.60557491030811541657963881620, 6.76913046733328886161902253858, 7.40982171108430778121626043824, 8.174594550583308790631073415644
