L(s) = 1 | − 2.50·2-s − 3-s + 4.29·4-s − 0.412·5-s + 2.50·6-s + 4.38·7-s − 5.75·8-s + 9-s + 1.03·10-s + 0.261·11-s − 4.29·12-s + 0.179·13-s − 10.9·14-s + 0.412·15-s + 5.83·16-s + 2.65·17-s − 2.50·18-s − 5.43·19-s − 1.77·20-s − 4.38·21-s − 0.655·22-s + 4.23·23-s + 5.75·24-s − 4.82·25-s − 0.451·26-s − 27-s + 18.8·28-s + ⋯ |
L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.14·4-s − 0.184·5-s + 1.02·6-s + 1.65·7-s − 2.03·8-s + 0.333·9-s + 0.327·10-s + 0.0787·11-s − 1.23·12-s + 0.0498·13-s − 2.93·14-s + 0.106·15-s + 1.45·16-s + 0.643·17-s − 0.591·18-s − 1.24·19-s − 0.396·20-s − 0.955·21-s − 0.139·22-s + 0.884·23-s + 1.17·24-s − 0.965·25-s − 0.0884·26-s − 0.192·27-s + 3.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5965711207\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5965711207\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 5 | \( 1 + 0.412T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 - 0.261T + 11T^{2} \) |
| 13 | \( 1 - 0.179T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 0.658T + 41T^{2} \) |
| 43 | \( 1 + 0.0316T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 + 4.71T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 - 3.55T + 79T^{2} \) |
| 83 | \( 1 - 8.28T + 83T^{2} \) |
| 89 | \( 1 - 3.47T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−10.86770567692972596559443710319, −10.34161036374082529071881344785, −9.217362223129506460807331773469, −8.313321539255942537028668369899, −7.77093971825569694591152831707, −6.82836823410667379345814207107, −5.64607999592221994554203270222, −4.35734341873061834267671373737, −2.22316602017455679442201881332, −1.01952513330390515912802301514,
1.01952513330390515912802301514, 2.22316602017455679442201881332, 4.35734341873061834267671373737, 5.64607999592221994554203270222, 6.82836823410667379345814207107, 7.77093971825569694591152831707, 8.313321539255942537028668369899, 9.217362223129506460807331773469, 10.34161036374082529071881344785, 10.86770567692972596559443710319